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

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

statistical tests optimally meeting certain fuzzy requirements on the power function and on the sample size abstract usually the parameters of statistical tests are determined by crisp constraints on the probabilities and of the errors of type i and of type ii respectively and by the additional condition that the sample size n is to be selected as small as possible in this paper this approach is modified here statistical tests are proposed to be evaluated by the degree to which certain fuzzy requirements on and n are met aggregating these requirements by the minimum operator a procedure is presented how to determine a test maximizing this degree of satisfaction statistical tests with a continuously distributed test statistic and in particular two sided gau tests are discussed in more detail

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

representation theorems for l fuzzy quantities abstract in this paper representation theorems are given for l fuzzy quantities which permit a better understanding of fuzziness in particular representation theorems due to negoita and ralescu 17 and to sherwood and taylor 23 are extended to the scope of complete and brouwerian lattices

testing fuzzy hypotheses with crisp data abstract in this paper an approach is presented how statistical tests originally constructed to examine crisp hypotheses can also be applied to fuzzily formulated hypotheses in particular criterions and are proposed generalizing the probabilities of the errors of type i and type ii respectively the general approach is applied to one and two sided gaus tests here diagrams are given to determine the critical values in the most popular cases of 0 01 and 0 05 if in addition the value of is fixed in advance the sample size of a one or two sided gaus test can be obtained using supplementary graphs

combining ambiguous evidence with respect to ambiguous a priori knowledge part ii fuzzy logic abstract this paper describes fuzzy conditioned dempster shafer fcds theory a probability based calculus for dealing with possibly imprecise and vague evidence when the underlying a priori knowledge base is itself possibly imprecise and vague fcds has the same general form as existing fuzzy dempster shafer theories fcds however employs a finite level zadeh fuzzy logic and a dempster shafer like combination operator which is conditioned to reflect the influence of any a priori knowledge base which can be modeled by a belief measure on finite level fuzzy sets we show that fcds is grounded in probability theory specifically in the theory of random fuzzy sets we also show that it is a generalization of bayesian theory to the case when both evidence and a priori knowledge are imprecise and vague we show that fcds is consistent with the lattice structure of zadeh s min max fuzzy logic and that in particular it possesses an analog of the familiar mobius transform

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

an overview of wavelet based multiresolution analyses in this paper an overview of wavelet based multiresolution analyses is presented first the continuous wavelet transform in its simplest form is discussed then the definition of a multiresolution analysis is given and how wavelets fit into it is shown the authors take a closer look at orthogonal biorthogonal and semiorthogonal wavelets the fast wavelet transform wavelets on an interval multidimensional wavelets and wavelet packets are discussed several examples of wavelet families are introduced and compared finally the essentials of two major applications are outlined data compression and compression of linear operators

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

dealing with uncertainties in engineering problems using only available data abstract this paper is the second in the series of three addressing the problem of how to deal with uncertainties in the modelings of engineering problems using only available data the problem of the heat exchanger was selected as the example because it is sufficiently complex but still simple so that the difficulties and ideas will be apparent there are various theories on how to treat the uncertainties in the first paper in the series we use only the fuzzy set respectively the possibility theory in this paper we use the fuzzy approach for some model parameters and the probability approach for the others the third paper will use the bayesian theory

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

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

fuzzy hypothesis testing with hybrid data abstract the formulation of binary hypothesis testing where the available data under one hypothesis is a superposition of a random and a fuzzy component is addressed in this formulation the likelihood ratio which is normally utilized in such types of decision making problems is fuzzified and compared to a threshold this involves the application of principles of ordering fuzzy sets which have been developed elsewhere by the authors decision rules are described which are independent of the shape of the membership function representing the fuzzy data these decision rules are illustrated for the case where the random component of the data is normally distributed probability of error performance curves using different ordering criteria are obtained and compared to the performance of non fuzzy hypothesis testing

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

inner and outer fuzzy approximations of confidence intervals we extend the notion of confidence region to fuzzy data by defining a pair of fuzzy inner and outer confidence regions we show the connection with previous proposals as well as with recent studies on hypothesis testing with low quality data

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

a fast algorithm for vortex blob interactions abstract we introduce a new fast summation a algorithm which unlike previous methods is not restricted to potential functions its asymptotic complexity is o nh d 1 where n is the number of blobs h is the level of refinement and d is the domain dimension for a nonsingular kernel the complexity reduces to o n we present a complete error and complexity analysis of the algorithm and describe its implementation as an example we use the vortex blob method to simulate vortex sheet motion in incompressible flow

aleatoric and epistemic uncertainty in groundwater flow and transport simulation 1 the characterization of aleatory hydrogeological parameter uncertainty has traditionally been accomplished using probability theory however when consideration is given to epistemic as well as aleatory uncertainty probability theory is not necessarily appropriate this is especially the case where expert opinion is regarded as a suitable source of information when experts opine upon the uncertainty of a parameter value both aleatoric and epistemic uncertainties are introduced and must be modeled appropriately a novel approach to expert provided parameter uncertainty characterization can be defined that bridges an historical gap between probability theory and fuzzy set theory herein a random set a generalization of a random variable is employed to formalize expert knowledge and fuzzy sets are used to propagate this uncertainty to model estimates of contaminant transport the resultant random set based concentration estimates are shown to be more general than the corresponding random variable estimates in some cases the random set based results are shown as upper and lower probabilities that bound the corresponding random variable s cumulative distribution function

fuzzy measures for a correlation coefficient of fuzzy numbers under tw the weakest t norm based fuzzy arithmetic operations recently the sup min convolution based on zadeh s extension principle has been used by liu and kao fuzzy measures for correlation coefficient of fuzzy numbers fuzzy sets and systems 128 2002 267 275 to calculate a fuzzy correlation coefficient they used a mathematical programming approach to derive fuzzy measures based on the classical definition of the correlation coefficient it is well known that t w the weakest t norm based addition and multiplication preserve the shape of l r fuzzy numbers in this paper we consider the computational aspect of the t w based extension principle when the principle is applied to a correlation coefficient of l r fuzzy numbers we give the exact solution of a fuzzy correlation coefficient without programming or the aid of computer resources

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

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

identification of weak nonlinearities on damping and stiffness by the continuous wavelet transform we consider the free response of a nonlinear vibrating system using the ridges and skeletons of the continuous wavelet transform we identify weak nonlinearities on damping and stiffness and estimate their physical parameters the crucial choice of the son wavelet function is obtained using an optimization technique based on the entropy of the continuous wavelet transform the method is applied to simulated single degree of freedom systems and multi degree of freedom systems with nonlinearities on damping and stiffness experimental validation of the nonlinear identification and parameter estimation method is presented the experimental system is a clamped beam with nonlinearities on damping and stiffness and these nonlinearities are identified and quantified from a displacement sensor

machine fault diagnosis through an effective exact wavelet analysis continuous wavelet transforms cwts are widely recognized as effective tools for vibration based machine fault diagnosis as cwts can detect both stationary and transitory signals however due to the problem of overlapping a large amount of redundant information exists in the results that are generated by cwts the appearance of overlapping can smear the spectral features and make the results very difficult to interpret for machine operators misinterpretation of results may lead to false alarms or failures to detect anomalous signals moreover as conventional cwts only use a single mother wavelet to generate daughter wavelets the distortion of the original signal in the resultant coefficients is inevitable obviously this will significantly affect the accuracy in anomalous signal detection to minimize the effect of overlapping and to enhance the accuracy of fault detection a novel wavelet transform which is named as exact wavelet analysis has been designed for use in vibration based machine fault diagnosis the design of exact wavelet analysis is based on genetic algorithms at each selected time frame the algorithms will generate an adaptive daughter wavelet to match the inspected signal as exactly as possible the optimization process of exact wavelet analysis is different from other adaptive wavelets as it considers both the optimization of wavelet coefficients and the satisfaction of the admissibility conditions of wavelets the results obtained from simulated and practical experiments prove that exact wavelet analysis not only minimizes the undesirable effect of overlapping but also helps operators to detect faults and distinguish the causes of faults with the help from exact wavelet analysis sudden shutdowns of production and services due to the fatal breakdown of machines could be avoided

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

an extension of draghicescu s fast tree code algorithm to the vortex method on a sphere a fast and accurate algorithm to compute interactions between n point vortices and between n vortex blobs on a sphere is proposed it is an extension of the fast tree code algorithm developed by draghicescu for the vortex method in the plane when we choose numerical parameters in the fast algorithm suitably the computational cost of o n 2 is reduced to o n logn 4 and the approximation error decreases like o 1 n when n as demonstrated in the present article we also apply the fast method to long time evolution of two vortex sheets on the sphere to see the efficiency a key point is to describe the equation of motion for the n points in the three dimensional cartesian coordinates

extension principles for fuzzy set theory in several cases we show that it is possible to extend a notion in classical mathematics by identifying each fuzzy subset with the continuous chain of its closed cuts and by applying this notion to these cuts in particular this idea is applied to extend functions from subsets into subsets for instance closure operators and functions from sets into real numbers for instance measures

kendall s rank correlation on quantized data an interval valued approach kendall s rank correlation coefficient also called kendall s is an efficient and robust way for identifying monotone relationships between two data sequences however when applied to digital data the high number of ties yields inconsistent results due to quantization here we propose an extension of kendall s that considers an epistemic view of a sequence of quantized data each sample is supposed to be the quantized version of an original value that is a real number we come up with an imprecise defined as the interval containing all values that could have been computed on sequences of original values before quantization we propose a very simple and straightforward algorithm to compute this interval valued we prove the exactness of the bounds and propose an experiment that illustrates the need for such an extension

upper and lower probabilities induced by a fuzzy random variable we review two existing interpretations of fuzzy random variables in the first one the fuzzy random variable is viewed as a linguistic random variable in the second case it represents some incomplete knowledge about an otherwise standard random variable for each interpretation the information provided by the frv is described by a specific model namely a standard probability model and a second order imprecise model respectively in this paper we deal with an alternative interpretation guided by simple examples we will observe the usefulness of each interpretation when applied to particular situations then we will demonstrate that the new interpretation leads in a natural way to the pair of order capacities furthermore we show that they are formally related to the former models these results can be applied in future works to make inferences from fuzzy sample data the use of upper lower models instead of second order models will enable us in the future to reach crisp decisions in some specific statistical problems without adding any arbitrary information and taking into account the imprecision in data

median value and median interval of a fuzzy number median value and median interval of a fuzzy number are investigated effects of concentration and dilation of a trapezoidal fuzzy number on the location of the median value and the median interval are studied relationship between the median interval and the mean interval is explored central value and central interval of a fuzzy number are proposed

statistical tests for fuzzy data abstract in this paper we investigate the impacts of vague and imprecise data on the statistical task of hypotheses testing test results are evaluated with respect to this ambiguity the issue of how to aggregate fuzzy data to a fuzzy sample vector is discussed and different eventualities are considered their consequences on the test results are compared the domain of the used test statistics is extended to fuzzy sample vectors by the means provided in the first part of this paper different methods for the defuzzification of the obtained fuzzy test results are discussed and the fuzzy nature of the classical verification task for statistical hypotheses is studied in this context a way of how to overcome the general insufficiency of statistical tests of being conclusive only in one direction is proposed by admitting fuzzy results

fuzzy hypotheses for guha implications abstract this paper presents a fuzzy generalization of a sophisticated approach to exploratory data analysis the general unary hypotheses automaton guha the guha paradigm to automatically generate sentences of an observational calculus which are supported by given data has been attracting attention for nearly 30 years most of the procedures for generating observational sentences encountered in the existing implementations of the approach are based on statistical hypotheses testing however there is an inner contradiction inherent to using common statistical tests to this end those tests always require a precise formulation of the tested conditions on the random variables characterizing the underlying real phenomena on the other hand due to its explorative purpose guha is predominantly used in situations in which the user has only a rather vague knowledge of those random variables thus being unable to precisely state the conditions to be tested therefore a fuzzy generalization of the approach is proposed it is based on a method of fuzzy hypotheses testing which was elaborated for the statistical tests used in guha and integrated into the context of an observational calculus due to space limitations the paper concentrates on a particular kind of generated observational sentences guha implications the fuzzy counterparts of two key concepts pertaining to guha implications are defined and a number of their properties are established the proposed approach is illustrated on two commonly used guha implications

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

three ways to solve the poisson equation on a sphere with gaussian forcing motivated by the needs of vortex methods we describe three different exact or approximate solutions to the poisson equation on the surface of a sphere when the forcing is a gaussian of the three dimensional distance 2 j exp 2 e 2 1 cos q c g a u s s e more precisely the forcing is a gaussian minus the gauss constraint constant c g a u s s this subtraction is necessary because j is bounded for any type of forcing only if the integral of the forcing over the sphere is zero y kimura h okamoto vortex on a sphere j phys soc jpn 56 1987 4203 4206 d g dritschel contour dynamics surgery on the sphere j comput phys 79 1988 477 483 the legendre polynomial series is simple and yields the exact value of the gauss constraint constant but converges slowly for large e the analytic solution involves nothing more exotic than the exponential integral but all four terms are singular at one or the other pole cancelling in pairs so that j is everywhere nice the method of matched asymptotic expansions yields simpler uniformly valid approximations as series of inverse even powers of e that converge very rapidly for the large values of e e 40 appropriate for geophysical vortex computations the series converges to a nonzero o exp 4 e 2 error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order

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

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

a study of defuzzification with experts knowledge for deteriorating repairable systems abstract this paper presents ideas on the applications of fuzzy concepts to decision making for deteriorating repairable systems a non homogeneous poisson process nhpp with a power law intensity function is used in this study in general classical bayesian decision methods presume that future states of nature can be characterized as probability events however we do not know what the future will entail probabilistically so we devise a method to consider experts opinions which are usually the absence of sharply defined criteria and to develop a fuzzy bayesian decision process for dealing with such situations two cases of the discrimination problem with deteriorating repairable systems are studied 1 fuzzy states and exact information and 2 fuzzy states and fuzzy information the fuzzy decomposition and arithmetic derivation of the experts opinion are presented to facilitate the development of the bayesian decision process for deteriorating repairable systems

similarity of personal preferences theoretical foundations and empirical analysis we study the problem of defining similarity measures on preferences from a decision theoretic point of view we propose a similarity measure called probabilistic distance that originates from the kendall s tau function a well known concept in the statistical literature we compare this measure to other existing similarity measures on preferences the key advantage of this measure is its extensibility to accommodate partial preferences and uncertainty we develop efficient methods to compute this measure exactly or approximately under all circumstances these methods make use of recent advances in the area of markov chain monte carlo simulation we discuss two applications of the probabilistic distance in the construction of the decision theoretic video advisor diva and in robustness analysis of a theory refinement technique for preference elicitation

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

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

a hierarchical o n force calculation algorithm a novel code for the approximate computation of long range forces between n mutually interacting bodies is presented the code is based on a hierarchical tree of cubic cells and features mutual cell cell interactions which are calculated via a cartesian taylor expansion in a symmetric way such that total momentum is conserved the code benefits from an improved and simple multipole acceptance criterion that reduces the force error and the computational effort for n 104 the computational costs are found empirically to rise sublinearly with n for applications in stellar dynamics this is the first competitive code with complexity o n it is faster than the standard tree code by a factor of 10 or more

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

approximate fuzzy arithmetic operations using monotonic interpolations abstract we suggest the use of piecewise monotonic interpolations to approximate and represent a fuzzy number or interval and to derive a procedure to control the absolute error associated to the arithmetic operations between fuzzy numbers in order to reduce the distance between the true result of the operation and its approximation the monotonic functions are then used to define a parametric representation of a large class of fuzzy numbers having general shapes of the membership function and a simple and accurate procedure is introduced for the fuzzy arithmetic several computational experiments are given to show the good performance of the proposed procedure

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

explicit parallel poisson integration of point vortices on the sphere point vortex models are frequently encountered in conceptual studies in geophysical fluid dynamics but also in practical applications for instance in aeronautics in spherical geometry the motion of vortex centres is governed by a dynamical system with a known poisson structure we construct poisson integration methods for these dynamics by splitting the hamiltonian into its constituent vortex pair terms from backward error analysis the method is formally known to provide solutions to a modified poisson system with the correct bracket but with a modified hamiltonian function different orderings of the pairwise interactions are considered and also used for the construction of higher order methods the energy and momentum conservation of the splitting schemes is demonstrated for several test cases for particular orderings of the pairwise interactions the schemes allow scalable parallelization this results in a linear as opposed to quadratic scaling of computation time with system size when scaling the number of processors accordingly

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

nonparametric kernel estimation based on fuzzy random variables this paper extends the classical nonparametric curve fitting methods for fuzzy random variables the study was conducted in two parts 1 fuzzy density estimation and 2 nonparametric regression the classical nonparametric density estimation was first developed based on the kernel method for a given fuzzy random sample at crisp and or fuzzy point based on crisp or fuzzy bandwidth the classical bandwidth selection was also extended when the underlying population was normal with known or unknown fuzzy variance moreover the classical nonparametric kernel based regression model with crisp and or input or output is extended whenever a given bandwidth is a crisp or fuzzy number the cross validation procedure for selecting the optimal value of the crisp smoothing parameter is also extended to fit the proposed nonparametric regression model the large sample properties of the proposed fuzzy estimators were also investigated by some theorems several numerical examples including that of real life data are used to illustrate the proposed methods in curve fitting estimation with crisp and or fuzzy information moreover the proposed methods were examined in comparison with some other existing methods and their effectiveness were clarified via some numerical examples and simulation studies both theatrical and numerical results indicated that the nonparametric curve fitting methods significantly reduced the sum of square errors as well as the spreads of the fuzzy curve fitting estimation

a natural way for improving the accuracy of the continuous wavelet transforms aiming at further increasing the accuracy of the continuous wavelet transforms cwts of transient vibration signals a new coefficient trimming technique is proposed in this paper different from the conventional cwt analysis which treats different scales of daughter wavelets equally and without using any discrimination the proposed technique takes the role of the daughter wavelet into account in the cwt calculations in the concept of this new technique not only different scales of daughter wavelets play different roles in the calculations but also the daughter wavelet at the same scale also plays a different role at different time instants the role of the daughter wavelet is numerically evaluated through the approach of the correlation between the daughter wavelet and the instant signal within the time sliding window which window length is predefined by using a compromise strategy once the correlation coefficient indicating the role of a daughter wavelet at certain time instant is obtained it will be applied to trim the corresponding cwt coefficient that is derived at the corresponding wavelet scale and the current time instant the details of this operation are given in the following context of the paper extensive experiments show that with the aid of this proposed technique the false information in the cwt map is successfully removed while the true information of the vibration signal is retained sufficiently moreover different from the threshold based purification methods the continuity and the smoothness of the wavelet coefficients are maintained perfectly in the whole process so the results obtained by using the proposed technique are more explicit and reliable from which the signal features may be identified more easily and correctly this is very helpful for promoting understanding of the dynamic behaviour of the machines and furthermore giving a more reliable assessment of their running condition

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

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

combined rank tests for randomly censored paired data abstract many authors have dealt with the problem of extending ordinary two sample rank tests to cases where censoring occurs albers and akritas 1987 proposed simple tests for this purpose based on the idea of making separate rankings for uncensored and censored observations and subsequently combining the resulting two rank statistics similarly some papers appeared that studied the problem of censoring for the paired data case rather than the two sample case this article indicates how the approach of albers and akritas can be adapted to the paired data case the tests involved are not based on the ranks of the differences but the differences of the ranks in the pooled sample in this way use is made of interblock information the asymptotic distribution of the new statistic is obtained under the null hypothesis and contiguous location alternatives by way of example wilcoxon and savage type scores are introduced that are optimal for logistic location and exponential scale alternatives respecti

approximated fuzzy lr computation this note addresses the problem of the multiplication and division of fuzzy numbers represented in lr left and right form it is known that usually the multiplication of lr fuzzy numbers does not provide lr fuzzy number however for real time applications for instance the complexity of such operation often prevents the use of fuzzy arithmetic framework to these fields this rises the question whether an approximated lr fuzzy number can be found in case of fuzzy multiplication division operation this paper focuses on an approximated lr fuzzy number that agrees with the result supplied by the fuzzy multiplication division operation according to some rational criteria these criteria involve properties of global specificity high plausibility and typicality representation furthermore the error induced by this approximation is evaluated and included as a constraint for the approximation issue

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

a two sample test with interval censored data via multiple imputation interval censored data arise naturally in large scale panel studies where subjects can only be followed periodically and the event of interest can only be recorded as having occurred between two examination times in this paper we consider the problem of comparing two interval censored samples we propose to impute exact failure times from interval censored observations to obtain right censored data then apply existing techniques such as harrington and fleming s g rho tests to imputed right censored data to appropriately account for variability a multiple imputation algorithm based on the approximate bayesian bootstrap abb is discussed through simulation studies we find that it performs well the advantage of our proposal is its simplicity to implement and adaptability to incorporate many existing two sample comparison techniques for right censored data the method is illustrated by reanalysing the breast cosmesis study data set

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

application of fuzzy naive bayes and a real valued genetic algorithm in identification of fuzzy model we present a method to identify a fuzzy model from data by using the fuzzy naive bayes and a real valued genetic algorithm the identification of a fuzzy model is comprised of the extraction of if then rules that is followed by the estimation of their parameters the involved parameters include those which determine the membership function of fuzzy sets and the certainty factors of fuzzy if then rules in our method as long as the fuzzy partition in the input output space is given the certainty factor of each rule is computed with the fuzzy conditional probability of the consequent conditioned on the antecedent by using the fuzzy naive bayes which is a generalization of naive bayes the fuzzy model involves the rules characterized by the highest values of certainty factors the certainty factor of each rule is the fuzzy conditional probability and it reflects the inner relationship between the antecedent and the consequent in order to improve the accuracy of the fuzzy model the real valued genetic algorithm is incorporated into our identification process this process concerns the optimization of the membership functions occurring in the rules we just involve the parameters of membership function of the fuzzy sets into the real valued genetic algorithm since the certainty factor of each rule can be computed automatically the performance of the model is shown for the backing truck problem and the prediction of mackey glass time series