Medical diagnosis: Fuzzy sets and degrees of belief

A model simulating the medical diagnostic process is presented. Diagnostic groups are considered as fuzzy sets. Partial knowledge is quantified by belief functions. The process is considered as the evaluation by the clinician of his degree of belief concerning the belonging of his patient to a fuzzy set given fuzzy and partial informations.     

A characterization of spherical distributions

It is shown that when the random vector X in Rⁿ has a mean and when the conditional expectation E(u′X|v′X) = 0 for all vectors u, v Rⁿ which satisfy u′v = 0, then the distribution of X is orthogonally invariant. A version of this characterization is also established when X does not have a mean vector.     

Approximations to the distributions of ordered distance random variables

Summary In the present note we give short proofs of asymptotic theorems for the distributions of extreme and intermediate ordered distance random variables. Moreover, a quick goodness-of-fit test is proposed which is based on a single intermediate ordered distance random variable.     

On a new system of discrete distributions and characterizations of several discrete distributions by equality of distributions

This paper deals with a new system of discrete distributions. It also gives several characterizations of the Waring (and hence the Yule) distribution (and its truncated versions), the super-Poisson, the discrete uniform and other discrete distributions by using this system and other such systems existing in the literature, and linear regression. Continuous analogues of the above results are also briefly discussed.     

A note on characterizations of multivariate stable distributions

Several characterizations of multivariate stable distributions together with a characterization of multivariate normal distributions and multivariate stable distributions with Cauchy marginals are given. These are related to some standard characterizations of marcinkiewicz.Research supported, in part, by the Air Force Office of Scientific Research under Contract AFOSR 84-0113. Reproduction in whole or part is permitted for any purpose of the United States Government.     

A taxonomy and review of the fuzzy data envelopment analysis literature: Two decades in the making

Data envelopment analysis (DEA) is a methodology for measuring the relative efficiencies of a set of decision making units (DMUs) that use multiple inputs to produce multiple outputs. Crisp input and output data are fundamentally indispensable in conventional DEA. However, the observed values of the input and output data in real-world problems are sometimes imprecise or vague. Many researchers have proposed various fuzzy methods for dealing with the imprecise and ambiguous data in DEA. In this study, we provide a taxonomy and review of the fuzzy DEA methods. We present a classification scheme with four primary categories, namely, the tolerance approach, the α-level based approach, the fuzzy ranking approach and the possibility approach. We discuss each classification scheme and group the fuzzy DEA papers published in the literature over the past 20 years. To the best of our knowledge, this paper appears to be the only review and complete source of references on fuzzy DEA.     

A result on the composition of distributions

LetF be a distribution and letf be a locally summable function. The distributionF(f) is defined as the neutrix limit of the sequenceF _n (f), whereF _n (x) = F(x) * δ _n (x) andδ _n (x) is a certain sequence of infinitely differentiable functions converging to the Dirac delta-functionδ(x). The distribution (x^r)^−s is valuated forr, s = 1,2, ….     

Some properties of subexponential distributions

The nonnegative random variableX is said to have a subexponential distribution if we have (1-G(t))/(1-F(t))→2 ast→∞, whereF(t)=P{X≤t} andG(t) is the convolution ofF(t) with itself. Conditions on the distribution of independent nonnegative random variablesX andY such that max(X, Y) and min(X, Y) have a subexponential distribution are given. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 138–144, July, 1997. Translated by N. K. Kulman     

Fuzzy system reliability analysis by the use of Tω (the weakest t-norm) on fuzzy number arithmetic operations

In general, the sup-min convolution has been used for fuzzy arithmetic to analyze fuzzy system reliability, where the reliability of each system component is represented by fuzzy numbers. It is well known that T_ω-based addition preserves the shape of L-R type fuzzy numbers. In this paper, we show T_ω-based multiplication also preserves the shape of L-R type fuzzy numbers. We then apply T_ω-based arithmetic operations to fuzzy system reliability analysis. In fact, we show that we can simplify fuzzy arithmetic operations and even get the exact solutions for L-R type fuzzy system reliability, while others [Singer, Fuzzy Sets Syst. 34 (1990) 145; Cheng and Mon, Fuzzy Sets Syst. 56 (1993) 29; Chen, Fuzzy Sets Syst. 64 (1994) 31] have got the approximate solutions using sup-min convolution for evaluating fuzzy system reliability.     

Fuzzy system reliability analysis by the use of Tω (the weakest t-norm) on fuzzy number arithmetic operations

In general, the sup-min convolution has been used for fuzzy arithmetic to analyze fuzzy system reliability, where the reliability of each system component is represented by fuzzy numbers. It is well known that T_ω-based addition preserves the shape of L-R type fuzzy numbers. In this paper, we show T_ω-based multiplication also preserves the shape of L-R type fuzzy numbers. We then apply T_ω-based arithmetic operations to fuzzy system reliability analysis. In fact, we show that we can simplify fuzzy arithmetic operations and even get the exact solutions for L-R type fuzzy system reliability, while others [Singer, Fuzzy Sets Syst. 34 (1990) 145; Cheng and Mon, Fuzzy Sets Syst. 56 (1993) 29; Chen, Fuzzy Sets Syst. 64 (1994) 31] have got the approximate solutions using sup-min convolution for evaluating fuzzy system reliability.     

A representation of the probability of a fuzzy subset

A representation of the probability associated with a fuzzy subset is presented. This representation, which is itself a fuzzy subset of the unit interval, is defined by the intersection of an upper and lower bound on the probability of the fuzzy event.     

An approach to spectral sets of certain zeta-functions

A. Beurling introduced the concept of spectral sets of unbounded functions to study the possibility of the approximation of those by trigonometric polynomials. The author studied spectral sets of the Riemann zeta-function and zeta-functions belonging to a certain class. The purpose of the present paper is to improve the results in the previous studies for spectral sets. The improvement is based on the relation between spectral sets and support of distributions.     

Asymptotic independence of distributions of normalized order statistics of the underlying probability measure

Let n denote the sample size, and let r_i ∈ {1,…,n} fulfill the conditions r_i ? r_i?1 ≥ 5 for i = 1,…,k. It is proved that the joint normalized distribution of the order statistics Z_ri:n, i = 1,…,k, is independent of the underlying probability measure up to a remainder term of order $\text{O((}\text{k}\text{n}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$. A counterexample shows that, as far as central order statistics are concerned, this remainder term is not of the order $\text{O((}\text{k}\text{n}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ if r_i ? r_i?1 = 1 for i = 2,…,k.     

Asymptotic distributions of the likelihood ratio test statistics for covariance structures of the complex multivariate normal distributions

In this paper, the authors derived asymptotic expressions for the null distributions of the likelihood ratio test statistics for multiple independence and multiple homogeneity of the covariance matrices when the underlying distributions are complex multivariate normal. Also, asymptotic expressions are obtained in the non-null cases for the likelihood ratio test statistics for independence of two sets of variables and the equality of two covariance matrices. The expressions obtained in this paper are in terms of beta series. In the null cases, the accuracy of the first terms alone is sufficient for many practical purposes.     

Joint distributions of random sets and their relation to copulas

Random sets are set-valued random variables. They have been applied in various fields like stochastic geometry, statistics, economics, engineering or computer science, and are often used for modeling uncertainty. In an earlier paper the author has defined joint capacity and joint containment functionals which are multivariate set functions describing the joint distribution of random sets. This paper is concerned with the question how copulas can be used to describe or model the dependence of random sets. It is demonstrated that a joint containment functional can be related to its margins by a family of copulas. Furthermore, the paper provides a first insight how copulas can be used to define joint containment functionals.     

On some contributions of John Horváth to the theory of distributions

Our main task is a presentation of J. Horváth's results concerning

•

singular and hypersingular integral operators,

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the analytic continuation of distribution-valued meromorphic functions, and

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a general definition of the convolution of distributions.

At some instances minor supplements to his results are given.     

A classification of the main probability distributions by minimizing the weighted logarithmic measure of deviation

The paper reanalyzes the following nonlinear program: Find the most similar probability distribution to a given reference measure subject to constraints expressed by mean values by minimizing the weighted logarithmic deviation. The main probability distributions are examined from this point of view and the results are summarized in a table.The author acknowledges the NSERC Canada Research Grant A-5712.     

Classifier fusion in the Dempster-Shafer framework using optimized t-norm based combination rules

When combining classifiers in the Dempster-Shafer framework, Dempster’s rule is generally used. However, this rule assumes the classifiers to be independent. This paper investigates the use of other operators for combining non independent classifiers, including the cautious rule and, more generally, t-norm based rules with behavior ranging between Dempster’s rule and the cautious rule. Two strategies are investigated for learning an optimal combination scheme, based on a parameterized family of t-norms. The first one learns a single rule by minimizing an error criterion. The second strategy is a two-step procedure, in which groups of classifiers with similar outputs are first identified using a clustering algorithm. Then, within- and between-cluster rules are determined by minimizing an error criterion. Experiments with various synthetic and real data sets demonstrate the effectiveness of both the single rule and two-step strategies. Overall, optimizing a single t-norm based rule yields better results than using a fixed rule, including Dempster’s rule, and the two-step strategy brings further improvements.