on theoretical pricing of options with fuzzy estimators in this paper we present an application of a new method of constructing fuzzy estimators for the parameters of a given probability distribution function using statistical data this application belongs to the financial field and especially to the section of financial engineering in financial markets there are great fluctuations thus the element of vagueness and uncertainty is frequent this application concerns theoretical pricing of options and in particular the black and scholes options pricing formula we make use of fuzzy estimators for the volatility of stock returns and we consider the stock price as a symmetric triangular fuzzy number furthermore we apply the black and scholes formula by using adaptive fuzzy numbers introduced by thiagarajah et al k thiagarajah s s appadoo a thavaneswaran option valuation model with adaptive fuzzy numbers computers and mathematics with applications 53 2007 831 841 for the stock price and the volatility and we replace the fuzzy volatility and the fuzzy stock price by possibilistic mean value we refer to both cases of call and put option prices according to the black scholes model and also analyze the results to greek parameters finally a numerical example is presented for both methods and a comparison is realized based on the results

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

on the variance of fuzzy random variables abstract this paper deals with an expectation and a real valued variance of fuzzy random variables the expectation and the variance of a fuzzy random variable is characterized by frechet s principle in a metric space we study properties of the variance of a fuzzy random variable and compare it with the common variance of real valued random variables using the expectation and the variance of fuzzy random variables we consider a linear regression problem and limit theorems

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

possibilistic decisions and fuzzy statistical tests the paper deals with the problem of the interpretation of the results of statistical tests in terms of the theory of possibility the well known concept in statistics of the observed test size p also known as p value and significance has been given a new possibilistic interpretation and generalised for the case of imprecisely defined statistical hypotheses and vague statistical data the proposed approach allows a practitioner to evaluate the test results using intuitive concepts of possibility necessity or indifference

on global identifiability for arbitrary model parametrizations it is a fundamental problem of identification to be able even before the data have been analyzed to decide if all the free parameters of a model structure can be uniquely recovered from data this is the issue of global identifiability in this contribution we show how global identifiability for an arbitrary model structure basically with analytic non linearities can be analyzed using concepts and algorithms from differential algebra it is shown how the question of global structural identifiability is reduced to the question of whether the given model structure can be rearranged as a linear regression an explicit algorithm to test this is also given furthermore the question of persistent excitation for the input can also be tested explicitly is a similar fashion the algorithms involved are very well suited for implementation in computer algebra one such implementation is also described

a fast algorithm to assess local structural identifiability the paper presents a novel method for assessing the local structural identifiability question for a general non linear state space model the method is a combination of i the application of a singular value decomposition to a parametric output sensitivity matrix that is created by simply integrating the model once and ii a symbolic computation for a reduced model that is guided by the svd results and allows a confirmation of the conclusions regarding identifiability obtained in the first step in case there is a lack of identifiability the symbolic computation quickly results in determination of the exact structure of the nullspace and a suitable re parametrisation the method is discussed in detail and applied to three case studies of which the last two are considerably large containing 22 and 43 parameters to be identified respectively

locally optimum detection in moving average non gaussian noise detection algorithms that are locally optimum bayes and also asymptotically optimum are developed for both coherent and incoherent signaling for arbitrary interference and signal waveforms when the dependence in the noise samples is represented by a moving average model this leads to receiver structures which are prewhitened versions of the locally optimum detectors in the independent case a probability of error expression in the ideal observer symmetric case the processing gain and the minimum detectable signal are derived in both cases these demonstrate by means of an expression comparing performance between this and the independent case that for the same large sample size n 1 an improvement in performance is always achieved when the noise samples are dependent without any additional complexity in receiver structure

the fuzzy hyperbolic inequality index associated with fuzzy random variables the aim of this paper is focussed on the quantification of the extent of the inequality associated with fuzzy valued random variables in general populations for this purpose the fuzzy hyperbolic inequality index associated with general fuzzy random variables is presented and a detailed discussion of some of the most valuable properties of this index extending those for classical inequality indices is given two examples illustrating the computation of the fuzzy inequality index are also considered some comments and suggestions are finally included

on the formalization of fuzzy random variables abstract in this paper we develop a discussion on the mathematical formalization of the concept of fuzzy random variable this discussion is mainly focused on finding an adequate notion of measurability to be coherent with the notions on the space these random elements take values

a fuzzy probabilistic information system comparison criterion applications and properties a phi fuzzy probabilistic information system comparison criterion is defined it is based on a nonnegative concave function considered over a fuzzy state space based on phi fuzzy probabilistic information system comparison criterion many interesting properties are studied connections with the classical criteria of lehmann and blackwell are made the function phi admits many interesting information theoretical particular cases these particular cases are divided in two parts shannon s and nonshannon s and are put in the last section

embedding problem of noncompact fuzzy number space e abstract in this paper we use the embedding j wu and zhang from noncompact fuzzy number space e into the concrete frechet space c 0 1 c 0 1 to discuss some properties of the e valued functions such as measurability integrability etc

new results for identifiability of nonlinear systems the parameter identifiability problem of deterministic non linear control systems is studied relations between nonlinear observability nonlinear functional expansions the uniqueness theorem of nonlinear realization theory and identifiability are investigated by using these relations necessary and sufficient conditions for identifiability are obtained for the first time these results provide insight into the identifiability problem for nonlinear systems

statistical management of fuzzy elements in random experiments part ii the fisher information associated with a fuzzy information system abstract in this paper we examine statistical management and analysis of fuzzy experimental data by quantifying the amount of information about the experiment that is contained in available data and using the quantification above to compare experiments and to determine the minimum increase in sample size to remove the loss of information due to fuzziness in the companion paper in this issue part 1 an interpretation of grades of membership as probabilities in statistical problems is presented thus after examining some different sources of fuzzy imprecision in random experiments we have concluded that the interpretation above could make only some sense when fuzziness is involved in the observer report anyway that interpretation could mean an intuitive but not a formal way to approach membership degrees in that particular case

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

locally optimum detection of signals in a generalized observation model the random signal case locally optimum detectors for weak random signals are derived for a generalized observation model incorporating signal dependent and multiplicative noise it is shown that the locally optimum random signal detectors in the generalized observation model are interesting generalizations of those that would be obtained in the purely additive noisy signal model examples of explicit results for the locally optimum detector test statistics are given for some typical cases both asymptotic and finite sample size performances of the locally optimum detectors are considered and compared with those of other standard detector structures

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

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 cartan dieudonne scherk theorems for complex s orthogonal matrices abstract let m n c be the set of all n by n matrices with complex entries let s n be the set of all nonsingular symmetric matrices in m n c let s n be the set of all nonsingular skew symmetric matrices in m n c and let s n s n s n let s s n be given an a m n c is called s orthogonal if a t s a s let o s be the set of all s orthogonal matrices in m n c an h o s is called a symmetry if rank h i 1 let h s be the set of all symmetries in o s we show that every q o s is a product of elements of h s if q i then q is a product of two elements of h s suppose that rank q i m 1 if s q i is not skew symmetric then q can be written as a product of m elements of h s and q cannot be written as a product of less than m elements of h s if s q i is skew symmetric and if s s n then q can be written as a product of m 2 elements of h s and q cannot be written as a product of less than m 2 elements of h s if s q i is skew symmetric and if s s n then q can be written as a product of m 1 elements of h s and q cannot be written as a product of less than m 1 elements of h s

fuzzy hypothesis testing for a population proportion based on set valued information abstract in this paper we propose fuzzy hypothesis testing for a proportion with crisp data as the exact generalized one tailed hypergeometric test with fuzzy non complementary hypotheses and incorporate the conventional hypergeometric test with crisp hypotheses into its framework in particular we formulate hypotheses as ontic or epistemic fuzzy sets since we assume that presumptions regarding the true population proportion can be expressed in the conjunctive or disjunctive reading further we suggest modeling of membership functions in terms of cost or frequency due to considerations of user s priorities or incomplete knowledge in relation to hypotheses formulation considering a hypothesis as union of its crisp and fuzzy areas we illustrate via real life examples that in contrast to classical test theory fuzzy hypothesis testing provides an additional partial and gradual consideration of the indifference zone for both complementary and non complementary hypotheses the test concept is introduced for both test methods a test of significance and an alternative test and leads to a crisp test decision considering further cost aspects like the sample size the generalized error criteria are derived and interpreted in relation to both hypotheses additionally a corresponding sensitivity analysis is presented for one of the most promising test types

on fuzzy bayesian inference abstract the paper combines methods from bayesian statistics with ideas from fuzzy set theory to generalize bayesian methods both for samples of fuzzy data and for prior distributions with non precise parameters applying the principle of propagation of fuzziness we introduce a fuzzy valued likelihood function bayes theorem for both fuzzy data and fuzzy priors a fuzzy bayes estimator fuzzy predictive densities and distributions and fuzzy h p d regions

bayesian reliability analysis for fuzzy lifetime data lifetime data are important in reliability analysis classical reliability estimation is based on precise lifetime data it is usually assumed that observed lifetime data are precise real numbers however some collected lifetime data might be imprecise and are represented in the form of fuzzy numbers thus it is necessary to generalize classical statistical estimation methods for real numbers to fuzzy numbers bayesian methods have proved to be very useful when the sample size is small there is little study on bayesian reliability estimation based on fuzzy lifetime data most of the reported works in this area is limited to single parameter lifetime distributions in this paper we propose a new method to determine the membership function of the estimates of the parameters and the reliability function of multi parameter lifetime distributions an artificial neural network is used to approximate the calculation process of parameter estimation and reliability prediction the genetic algorithm is used to find the boundary values of the membership function of the estimate of interest at any cut level this method can be used to determine the membership functions of the bayesian estimates of multi parameter distributions the effectiveness of the proposed method is illustrated with normal and weibull distributions

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

constructing membership functions using statistical data abstract determination of the membership functions is vital in practical applications of the fuzzy set theory this paper presents a guideline to construct the membership functions for fuzzy sets whose elements have a defining feature with a known probability density function pdf in the universe of discourse the method finds the smallest fuzzy set which assigns high average membership values to those objects with the defining features distributed according to the given pdf it is show that for any pdf the method is capable of generating membership functions in accordance with the possibility probability consistency principle membership functions derived from some of the well known pdfs and an application in solving noise contaminated linear system of equations are presented

parameter identifiability of nonlinear systems the role of initial conditions identifiability is a fundamental prerequisite for model identification it concerns uniqueness of the model parameters determined from the input output data under ideal conditions of noise free observations and error free model structure in the late 1980s concepts of differential algebra have been introduced in control and system theory recently differential algebra tools have been applied to study the identifiability of dynamic systems described by polynomial equations these methods all exploit the characteristic set of the differential ideal generated by the polynomials defining the system in this paper it will be shown that the identifiability test procedures based on differential algebra may fail for systems which are started at specific initial conditions and that this problem is strictly related to the accessibility of the system from the given initial conditions in particular when the system is not accessible from the given initial conditions the ideal i having as generators the polynomials defining the dynamic system may not correctly describe the manifold of the solution in this case a new ideal that includes all differential polynomials vanishing at the solution of the dynamic system started from the initial conditions should be calculated an identifiability test is proposed which works under certain technical hypothesis also for systems with specific initial conditions

a note on the operativeness of neyman pearson tests with fuzzy information abstract in a previous paper we have studied the extension of the neyman pearson optimality criterion of testing statistical hypotheses when the available experimental information involves fuzzy imprecision this extension usually becomes unmanageable for small samples nevertheless we will verify that when the sample size is large enough the application of the central limit theorem determines an operative extension

fuzzy hypotheses testing in the framework of fuzzy logic abstract testing hypotheses about the probability distribution underlying the available empirical data is one of the fundamental data analytic tasks in any application domain basically it consists in checking the null hypothesis that the probability distribution a priori assumed to belong to a certain set of distributions actually belongs to some its narrow subset which must be precisely delimited in advance however sometimes there are not enough clues for such a precise delimitation especially if the purpose of the data analysis is explorative a situation encountered increasingly often due to the growing amount of routinely collected data and the increasing importance of data mining that is why generalizations of statistical hypotheses testing to vague hypotheses have been investigated for more than a decade so far based on the most straightforward approach to replace the set defining the null hypothesis by a fuzzy set in this paper a principally different approach is presented motivated by the observational logic and its success in automated knowledge discovery its key idea is to view statistical testing of a fuzzy hypothesis as the application of an appropriate generalized quantifier of a fuzzy predicate calculus to predicates describing the data the theoretical principles of the approach are explained and its first implementations are briefly sketched

practical identifiability analysis of large environmental simulation models large environmental simulation models are usually overparameterized with respect to given sets of observations this results in poorly identifiable or nonidentifiable model parameters for small models plots of sensitivity functions have proven to be useful for the analysis of parameter identifiability for models with many parameters however near linear dependence of sensitivity functions can no longer be assessed graphically in this paper a systematic approach for tackling the parameter identifiability problem of large models based on local sensitivity analysis is presented the calculation of two identifiability measures that are easy to handle and interpret is suggested the first accounts for the sensitivity of model results to single parameters and the second accounts for the degree of near linear dependence of sensitivity functions of parameter subsets it is shown how these measures provide identifiability diagnosis for parameter subsets how they are able to guide the selection of identifiable parameter subsets for parameter estimation and how they facilitate the interpretation of the correlation matrix of the parameter estimate with respect to parameter identifiability in addition we show how potential bias of the parameter estimates due to a priori fixing of some of the parameters can be analyzed finally two case studies are presented in order to illustrate the suggested approach

fuzzy estimation for process capability indices process capability indices are summary statistics which measure the actual or the potential performance of process characteristics relative to the target and specification limits in most traditional methods precise estimation is used to assess the capability of manufacturing processes in this paper we introduce an algorithm based on buckley s estimation approach and use a family of confidence intervals to estimate process capability indices c p c p k and c p m the estimators of these indices thus obtained are triangular shaped fuzzy numbers we also present and illustrate method for the comparison of estimated process capability indices numerical examples are given to show the performance of the method

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

testing statistical hypotheses with vague data a definition of fuzzy test for testing statistical hypotheses with vague data is proposed then the general method for the construction of fuzzy tests for hypotheses concerning an unknown parameter against one sided or two sided alternative hypotheses is shown this fuzzy test contrary to the classical approach leads not to the binary decision to reject or to accept given null hypothesis but to a fuzzy decision showing a grade of acceptability of the null and the alternative hypothesis respectively however it is a natural generalization of the traditional test i e if the data are precise not vague we get a classical statistical test with the binary decision a measure of fuzziness of the considered fuzzy test is suggested and the robustness of that test is also discussed

on strong laws of large numbers for random upper semicontinuous functions a short proof of the strong law of large numbers for random upper semicontinuous functions fuzzy random variables with compact support is given this approach allows us to deduce many results on convergence of fuzzy random variables from the relevant results on random sets that appear as their level sets

fuzzy portfolio allocation models through a new risk measure and fuzzy sharpe ratio a new portfolio risk measure that is the uncertainty of portfolio fuzzy return is introduced in this paper beyond the well known sharpe ratio i e the reward to variability ratio in modern portfolio theory we initiate the so called fuzzy sharpe ratio in the fuzzy modeling context in addition to the introduction of the new risk measure we also put forward the reward to uncertainty ratio to assess the portfolio performance in fuzzy modeling corresponding to two approaches based on rm t rm m and rm t rm w fuzzy arithmetic two portfolio optimization models are formulated in which the uncertainty of portfolio fuzzy returns is minimized while the fuzzy sharpe ratio is maximized these models are solved by the fuzzy approach or by the genetic algorithm ga solutions of the two proposed models are shown to be dominant in terms of portfolio return uncertainty compared with those of the conventional mean variance optimization mvo model used prevalently in the financial literature in terms of portfolio performance evaluated by the fuzzy sharpe ratio and the reward to uncertainty ratio the model using rm t rm w fuzzy arithmetic results in higher performance portfolios than those obtained by both the mvo and the fuzzy model which employs rm t rm m fuzzy arithmetic we also find that using the fuzzy approach for solving multiobjective problems appears to achieve more optimal solutions than using ga although ga can offer a series of well diversified portfolio solutions diagrammed in a pareto frontier

the bayesian bootstrap the bayesian bootstrap is the bayesian analogue of the bootstrap instead of simulating the sampling distribution of a statistic estimating a parameter the bayesian bootstrap simulates the posterior distribution of the parameter operationally and inferentially the methods are quite similar because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions neither method should be applied without some consideration of the reasonableness of these model assumptions in this sense neither method is a true bootstrap procedure yielding inferences unaided by external assumptions

g reflectors analogues of householder transformations in scalar product spaces we characterize the analogues of householder transformations in matrix groups associated with scalar products and precisely delimit their mapping capabilities given a matrix group image and vectors x y necessary and sufficient conditions are derived for the existence of a householder like analogue image such that gx y when g exists we show how it can be constructed from x and y examples of matrix groups to which these results apply include the symplectic and pseudo unitary groups

constructive ways for generating generalized real orthogonal matrices as products of generalized symmetries the cartan dieudonne scherk cds theorem of algebraic group theory asserts that for fields of characteristic different from 2 every generalized orthogonal matrix can be written as the product of a certain minimal number of generalized symmetries when computing over r or c these symmetries are called householder matrices in numerical linear algebra they play an important role in orthogonal elimination and the qr factorization of matrices this note gives constructive proofs of a number of results to generate a generalized real or complex orthogonal or unitary matrix as the product of generalized householder matrices the proofs are patterned after the standard householder elimination practises of numerical analysis the original proofs of the cds theorem of around 50 years ago were not constructive but relied on the nonexistence of certain isotropic subspaces with respect to the underlying generalized inner product

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

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

performance characteristics of the fuzzy sign detector in this paper fuzzy statistical techniques are introduced into the optimum signal detection problems the likelihood ratio for fuzzy detection of known signals is obtained as a special case of the fuzzy signal detector a fuzzy set theoretic approach to sign detection of known signals is considered specifically the test statistic of the fuzzy sign detector for known signals is obtained some properties of the fuzzy sign nonlinearity which constitutes the fuzzy sign detector are also described finally the performance characteristics of the fuzzy sign detector are investigated and compared to those of the crisp sign detector

statistical hypotheses testing for fuzzy data the method of handling fuzziness upon the usual method of statistical hypotheses testing is proposed we transact the h level set of fuzzy data for the sake of invoking usual method of statistical hypotheses testing we propose the decision rules that are used to accept or reject the null and alternative hypotheses with the notions of degrees of optimism and pessimism by solving optimization problems we also provide a computational procedure and an example to clarify the discussions in this paper

the ϕs polar decomposition let sn s mnr s2 i sn s mnr s2 i and let sn sn sn for s sn let s mnc mnc be given by sa s 1ats an a mnc is called s symmetric if sa a a is called s skew symmetric if sa a and a is called s orthogonal if a is nonsingular and sa a 1 we say that a has a s polar decomposition if a qt for some s orthogonal q mnc and some s symmetric t mnc let vs a mnc s sa a we show that every nonsingular a vs has a s polar decomposition we determine conditions for which an a mnc has a s polar decomposition we also determine the possible jordan canonical forms of a s orthogonal matrix and of a s skew symmetric matrix

a generalized strong law of large numbers strong laws of large numbers have been stated in the literature for measurable functions taking on values on different spaces in this paper a strong law of large numbers which generalizes some previous ones like those for real valued random variables and compact random sets is established this law is an example of a strong law of large numbers for borel measurable nonseparably valued elements of a metric space

the subspaces spanned by householder vectors associated with an orthogonal or a symplectic matrix abstract the cartan dieudonne scherk theorem guarantees that every complex orthogonal matrix can be written as a product of matrices of the form h s u i u u t s where s i and u c n satisfies u t u 2 moreover every complex symplectic matrix can be written as a product of matrices of the form h s u i u u t s where s j 0 i i 0 and u 0 let a nonempty v c n be given the s orthogonal complement of v is v s z c n w t s z 0 for all w v the image of an n by n complex matrix a is the set of all z c n for which there is an x c n such that z a x and is denoted by im a let s i or s j suppose that q h s u 1 h s u 2 h s u r set u span u 1 u 2 u r we study the relationship between q u and im q i suppose that r is minimal we show that if dim u r then im q i u we also show that s q i is not skew symmetric if and only if dim u r let w im q i we show that a relationship between w and w s determines the jordan structure of q in particular we show that q i 2 0 if and only if w w s

fuzzy sets and statistical data abstract specific features of probability and possibility theories are discussed with emphasis on semantical aspects instead of putting forward the acknowledged usefulness of possibility theory for the non statistical modelling of subjective categories we try to figure out how statistical data and possibility theory could be matched as a result procedures for constructing weak possibilistic substitutes of probability measures and for processing imprecise statistical data are outlined they provide new insights on relationship between fuzzy sets and probability theories

advocating the use of imprecisely observed data in genetic fuzzy systems in our opinion and in accordance with current literature the precise contribution of genetic fuzzy systems to the corpus of the machine learning theory has not been clearly stated yet in particular we question the existence of a set of problems for which the use of fuzzy rules in combination with genetic algorithms produces more robust models or classifiers that are inherently better than those arising from the bayesian point of view we will show that this set of problems actually exists and comprises interval and fuzzy valued datasets but it is not being exploited current genetic fuzzy classifiers deal with crisp classification problems where the role of fuzzy sets is reduced to give a parametric definition of a set of discriminant functions with a convenient linguistic interpretation provided that the customary use of fuzzy sets in statistics is vague data we propose to test genetic fuzzy classifiers over imprecisely measured data and design experiments well suited to these problems the same can be said about genetic fuzzy models the use of a scalar fitness function assumes crisp data where fuzzy models a priori do not have advantages over statistical regression

statistical estimation with linguistic data abstract suppose the distribution of a population is characterized by a parameter whose value is not numerical as usually is the case but linguistic the problem is to make a good guess about especially in the case when only linguistic data are available in this paper the theory of fuzzy random variables is used to solve this problem a method for the construction of consistent and unbiased estimates is given

asymptotic properties of least squares estimation with fuzzy observations this paper deals with the asymptotic properties of the least squares estimators for fuzzy linear regression models with fuzzy triangular input output and random error terms the asymptotic normality and strong consistency of the fuzzy least squares estimator flse are investigated a confidence region based on a class of flses is proposed the asymptotic relative efficiency of flses with respect to the crisp least squares estimators is also provided and a numerical example is given some simulation results are also presented to illustrate the behavior of flses

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 fuzzy statistical test of fuzzy hypotheses abstract a fuzzy hypothesis is introduced and a testing method of a fuzzy hypothesis is proposed for random data the proposed test produces a conclusion which is also fuzzy some applications of the fuzzy statistical test are demonstrated particularly the test of the difference of two population means is considered in detail and the fuzzy hypothesis that two population means are nearly equal is tested by the proposed procedure

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

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

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

c fuzzy numbers and a dual of extension principle in this paper we introduce and investigate the concept of c fuzzy numbers we extend the algebraic operations on c fuzzy numbers and specially study the properties of these operations on lr type c fuzzy numbers in addition a dual of extension principle is introduced it is shown that the algebraic operations with c fuzzy numbers have a representation based on the dual of extension principle

equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems the problem of parameter identifiability has been considered from different points of view in the case of nonlinear dynamical systems for analytic systems the standard approach for uncontrolled systems is the taylor series approach pohjanpalo math biosciences 41 1978 21 or the approaches based on differential algebra for polynomial and rational systems the similarity transformation approach based on the local state isomorphism theorem gives a sufficient and necessary condition for global identifiability of nonlinear controlled systems but it leads only to a necessary condition for identifiability in the case of some uncontrolled systems our contribution consists in using the equivalence of systems based on the straightening out theorem to analyse the identifiability of uncontrolled systems from this theory we state the necessary or sufficient identifiability conditions some of them depending on the state variable dimension

two sample hypothesis tests of means of a fuzzy random variable abstract in this paper we will consider some two sample hypothesis tests for means concerning a fuzzy random variable in two populations for this purpose we will make use of a generalized metric for fuzzy numbers and we will develop an exact study for the case of normal fuzzy random variables and an asymptotic study for the case of simple general fuzzy random variables

ψs orthogonal matrices and ψs symmetries abstract let g l n c denote the set of n by n nonsingular matrices with entries from the field c of complex numbers for any s g l n c define the map s g l n c g l n c by s a s a 1 s 1 a matrix a g l n c is said to be s orthogonal if s a a 1 a s orthogonal h is called a s symmetry if rank h i 1 we give conditions on s so that s symmetries exist moreover we determine conditions on s such that the s symmetries generate the s orthogonal matrices and the minimum number of s symmetries needed to express a s orthogonal

connections between some criteria to compare fuzzy information systems abstract in previous papers we have examined the extension of some methods to compare random experiments to the case in which the available experiment information is fuzzy in this paper we analyze the relationships between the criteria based on blackwell s sufficiency and some criteria based on well known information measures

on statistically inference for fuzzy data with applications to descriptive statistics abstract the object of this paper is to discuss how fuzziness of data is propagated when statistical inference for samples of non precise data is carried out first fuzzy data and fuzzy samples are defined subsequently the method of propagation of fuzziness as used in previous work for specific stochastic models is formulated in rather general terms this method may be applied to any statistical method which leads to a result that may be expressed as a function f x 1 x n of the data x 1 x n it is proved that this method is equal to determining the images of a family of compact subsets of the sample space under the function f this equivalence is utilized in order to perform propagation of fuzziness in practice finally this approach is applied to some concepts of descriptive statistics such as fuzzy sample mean fuzzy sample variance and to the empirical distribution function of a fuzzy sample

a sequential selection method of a fixed number of fuzzy information systems based on the information energy gain on the basis of the information energy gain for fuzzy information systems a sequential selection method of a fixed number of fuzzy information systems is presented the information energy gain is a measure of the quantity of information concerning the state space provided by a fuzzy information system x when a prior probability distribution has been defined on the state space

an extensive form analysis for comparing fuzzy information systems by means of the worth and quietness of information abstract statistical decision problems deal with the choice among several actions whose consequences depend upon the state of nature and this choice is usually made on the basis of the outcome of an experiment selected from a collection of experiments whose probability measures depend upon the state the selection criteria among the experiments are generally based on some measures of the information contained in each experiment about that true state in this paper we suggest a new selection criterion among the experiments associated with a statistical decision problem when the experimental outcomes cannot be exactly perceived by the decision maker but rather the available information from each experimental outcome can be regarded as an element in a fuzzy information system as defined by tanaka okuda and asai in a previous paper we have proposed a method for comparing fuzzy information systems on the basis of the maximization of the oworth of information of a fuzzy park information system tanaka et al when the application of such a method leads to indifference between two fuzzy information systems this indifference can be avoided by applying the criterion in this paper this selection criterion is an extensive form analysis which is based on two measures of the information contained in a fuzzy information system about the true state the worth of information of a fuzzy information system and the expected quietness of information of a fuzzy information system the suitability of the criterion stated above is corroborated by studying its main properties and contrasting this procedure with other ones making use of different measures of information

a unified approach to fuzzy random variables the concept of fuzzy random variable was introduced as an analogous notion to random variables in order to extend statistical analysis to situations when the outcomes of some random experiment are fuzzy sets but in contrary to the classical statistical methods no unique definition has been established yet in this paper a set theoretical concept of fuzzy random variable will be presented this notion provides a useful framework to compare different concepts of fuzzy random variables using methods of general topology and drawing on results from topological measure theory and the theory of analytic spaces as the main result it will be shown that the introduced concept of fuzzy random variable is a unification of the already known ones

optimum and suboptimum space diversity detection of weak signals in non gaussian noise the structure of the locally optimum space diversity detector for coherent detection of a bandpass signal subject to amplitude fluctuations in the presence of narrow band non gaussian noise is derived and its large sample performance is evaluated for easy implementation a suboptimum structure is proposed it consists of local detectors performing independent local decisions and of a central processor that working according to simple logical schemes or or and performs the global decision based on the local ones numerical results and a comparison between the performances of both proposed structures are presented and discussed

approximating integrably bounded fuzzy random variables in terms of the generalized hausdorff metric abstract consider the class of the mappings from a euclidean space to the unit interval 0 1 that is the class of the fuzzy sets of this space which are upper semicontinuous the closure of their supports is compact and the inverse images of the singleton are nonempty sets when the metric d which generalizes the hausdorff metric is defined on the preceding class the resulting metric space is complete but nonseparable consequently for this nonseparability given a probability space and an integrably bounded fuzzy random variable associated with it one cannot ensure the existence of a sequence of simple fuzzy random variables d converging almost surely to the former one however we are going to show in this paper that for any given integrably bounded fuzzy random variable we can guarantee the existence of a sequence of integrably fuzzy random variables whose level functions are simple random compact sets and which d converges at every point to the former variable on the basis of this result we will obtain in this paper a constructive definition of integrably bounded fuzzy random variables and an operational way to compute their fuzzy expected values

a new fuzzy arithmetic in this paper we first present fuzzy numbers with a new parametric form based on this representation a new fuzzy arithmetic is defined and applied to fuzzy linear equations and fuzzy calculus

the λs householder matrices abstract let a s m n c be given suppose that s is nonsingular and hermitian then a is s orthogonal if a sa s let u c n be such that u su 0 the s householder matrix of u is s u i tuu s where t 2 u su we show that det s u 1 so that products of s householder matrices have determinant 1 let n 2 and let k be positive integers with k n set l k i k i n k we show that every l k orthogonal matrix having determinant 1 can be written as a product of at most 2 n 2 l k householder matrices we also determine the possible jordan canonical forms of products of two l k householder matrices

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

s orthogonal matrices and s symmetries abstract let s m n c be nonsingular a q m n c is called s orthogonal if q t s q s an s orthogonal h is called an s symmetry if rank h i 1 we give conditions on s so that every s orthogonal can be written as a product of s symmetries we give conditions on s so that some s orthogonal cannot be written as a product of s symmetries moreover we determine conditions on s so that there are no s symmetries

fuzzy least squares abstract several models for simple least squares fitting of fuzzy valued data are developed criteria are given for when fuzzy data sets can be fitted to the models and analogues of the normal equations are derived

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

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

partitions and fuzziness abstract we study classical or generalized partitions of a given finite set from two points of view on the one hand we consider fuzzy binary relations r defined on a finite set and transitive for a binary operation such that d 1 r is a distance we characterize such relations by matricial properties and we give several examples such as m supermetricity which is a necessary condition for m hypermetricity partitions are then constructed by means of cliques and spheres associated with the distance d the case of fuzzy partitions is specially investigated and an example from coding theory is given quasi partitions are connected with k ultrametricity on the other hand we characterize fuzzy partitions with the help of an information associated with a given threshold which is the lower value of the membership functions defining the fuzzy classes of by evaluating the probabilities of these classes we construct a quantity which increases when the partition is refined the gain of information resulting from the replacement of a given fuzzy partition by another one decreases when we sharpen a fuzzy partition and it measures the spread of it is then a good tool to choose a fuzzy partition in a family or to improve a given classification of the studied population for example

some statistical applications of generalized jensen difference divergence measures for fuzzy information systems in previous papers we have compared statistical experiments by considering unified scalar parametric generalizations of the jensen difference divergence measure in this paper we first introduce generalized jensen difference divergence measures when the available information from the experiment is not exact in order to compare fuzzy information systems then we establish the relationship between extended fisher information and the generalized r s jensen difference divergence measures

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

point estimation for the n sizes of random sample with one vague data abstract for a probability space a p a r v x r and random samples x 1 x 2 x n with one vague data associated with p d f f x 0 we tried to define sufficient statistic and unbiased estimator for them and also to test simple and likelihood ratio hypotheses with one vague data we have shown in our examples that using all the data including the vague data we can have the same result as using the n 1 stable data to test the hypotheses in crisp statistics alternatively if the result of our study can be generalized to several vague data it can be seen that we may not need to determine the outliers from our statistical data in the future