An axiomatic derivation of the coding-theoretic possibilistic entropy

We re-take the possibilistic (strictly non-probabilistic) model for information sources and information coding put forward in (Fuzzy Sets and Systems 132–1 (2002) 11–32); the coding-theoretic possibilistic entropy is defined there as the asymptotic rate of compression codes, which are optimal with respect to a possibilistic (not probabilistic) criterion. By proving a uniqueness theorem, in this paper we provide also an axiomatic derivation for such a possibilistic entropy, and so are able to support its use as an adequate measure of non-specificity, or rather of “possibilistic ignorance”, as we shall prefer to say. We compare our possibilistic entropy with two well-known measures of non-specificity: Hartley measure as found in set theory and U-uncertainty as found in possibility theory. The comparison allows us to show that the latter possesses also a coding-theoretic meaning.     

On the Evaluation of Uncertain Courses of Action

We consider the problem of decision making under uncertainty. The fuzzy measure is introduced as a general way of representing available information about the uncertainty. It is noted that generally in uncertain environments the problem of comparing alternative courses of action is difficult because of the multiplicity of possible outcomes for any action. One approach is to convert this multiplicity of possible of outcomes associated with an alternative into a single value using a valuation function. We describe various ways of providing a valuation function when the uncertainty is represented using a fuzzy measure. We then specialize these valuation functions to the cases of probabilistic and possibilistic uncertainty.     

Uniqueness of information measure in the theory of evidence

An evidence distribution on a set X assigns non-negative weights to the subsets of X. Such weights must sum to one and the empty set is given weight 0. An information measure can be defined for such an evidence distribution.If m_i are the weights assigned to subsets A_i, and a_i are the cardinalities of these subsets, then the function. Σ m_i log a_i satisfies all the usual axioms of an information measure. In this paper we show that, conversely, these axioms are sufficient to characterize uniquely the above measure. It can be thus considered as the main uncertainty function for the theory of evidence.We demonstrate that using only the properties of symmetry, additivity and subadditivity the problem of uniqueness can be reduced to finding linear functionals on the space of functions analytic at origin. We surmise that under a suitable continuity hypothesis, all such functionals can be represented as linear combinations of the coefficients of Taylor series. Our function then represents the first derivative evaluated at 0.An alternative approach is then discussed. We assume a form of branching property, suggested by the monotonicity considerations. Now the properties of symmetry, additivity and subadditivity, together with branching again offer the unique characterization of the information function. No continuity assumption whatsoever is needed and the proof is entirely elementary.     

A decision theory for partially consonant belief functions

Partially consonant belief functions (pcb), studied by Walley, are the only class of Dempster-Shafer belief functions that are consistent with the likelihood principle of statistics. Structurally, the set of foci of a pcb is partitioned into non-overlapping groups and within each group, foci are nested. The pcb class includes both probability function and Zadeh’s possibility function as special cases. This paper studies decision making under uncertainty described by pcb. We prove a representation theorem for preference relation over pcb lotteries to satisfy an axiomatic system that is similar in spirit to von Neumann and Morgenstern’s axioms of the linear utility theory. The closed-form expression of utility of a pcb lottery is a combination of linear utility for probabilistic lottery and two-component (binary) utility for possibilistic lottery. In our model, the uncertainty information, risk attitude and ambiguity attitude are separately represented. A tractable technique to extract ambiguity attitude from a decision maker behavior is also discussed.     

Probabilistic rough set over two universes and rough entropy

In this paper, we discuss the properties of the probabilistic rough set over two universes in detail. We present the parameter dependence or the continuous of the lower and upper approximations on parameters for probabilistic rough set over two universes. We also investigate some properties of the uncertainty measure, i.e., the rough degree and the precision, for probabilistic rough set over two universes. Meanwhile, we point out the limitation of the uncertainty measure for the traditional method and then define the general Shannon entropy of covering-based on universe. Then we discuss the uncertainty measure of the knowledge granularity and rough entropy for probabilistic rough set over two universes by the proposed concept. Finally, the validity of the methods and conclusions is tested by a numerical example.     

Measuring the Student Group Capacity for Obtaining Geometric Information in the van Hiele Developmental Thought Process：A Fuzzy Approach

1　 IntroductionThe van Hiele level theory of geometric reasoning[1,5,9] describes the ways of studentreasoning inEuclidean geometry.The ontogeny of thiscognitive activity of mental developmentis characterised byfive hierarchical and qualitatively differe…     

A measure based approach to the fusion of possibilistic and probabilistic uncertainty

We are interested in the problem of multi-source information fusion in the case when the information provided has some uncertainty. We note that sensor provided information generally has a probabilistic type of uncertainty whereas linguistic information typically introduces a possibilistic type of uncertainty. More generally, we are faced with a problem in which we must fuse information with different types of uncertainty. In order to provide a unified framework for the representation of these different types of uncertain information we use a set measure approach for the representation of uncertain information. We discuss a set measure representation of uncertain information. In the multi-source fusion problem, in addition to having a collection of pieces of information that must be fused, we need to have some expert provided instructions on how to fuse these pieces of information. Generally these instructions can involve a combination of linguistically and mathematically expressed directions. In the course of this work we begin to consider the fundamental task of how to translate these instructions into formal operations that can be applied to our information. This requires us to investigate the important problem of the aggregation of set measures.     

Possibilistic nested logic programs and strong equivalence

In this paper, the class of possibilistic nested logic programs is introduced. These possibilistic logic programs allow us to use nested expressions in the bodies and heads of their rules. By considering a possibilistic nested logic program as a possibilistic theory, a construction of a possibilistic logic programing semantics based on answer sets for nested logic programs and the proof theory of possibilistic logic is defined. In order to define a general method for computing the possibilistic answer sets of a possibilistic nested program, the idea of equivalence between possibilistic nested programs is explored. By considering properties of equivalence between possibilistic programs, a process of transforming a possibilistic nested logic program into a possibilistic disjunctive logic program is defined. Given that our approach is an extension of answer set programming, we also explore the concept of strong equivalence between possibilistic nested logic programs. To this end, we introduce the concept of poss SE-models. Therefore, we show that two possibilistic nested logic programs are strong equivalents whenever they have the same poss SE-models.The expressiveness of the possibilistic nested logic programs is illustrated by a scenario from the medical domain. In particular, we exemplify how possibilistic nested logic programs are expressive enough for capturing medical guidelines which are pervaded by vagueness and qualitative information.     

The Information Content of Data in Defined Choice-Situations

This paper is concerned with what people's decisions imply about the usefulness of the data they receive. It does not deal with what might improve decision-making but with an attempt to understand what the measurement of information for decision-making involves. Operational definitions and measures of the information content of data are discussed. Ackoff and Emery's treatment of information as dependent on the choice-situation is followed, but a new concept, that of information inherent in the structure of the choice-situation is added. It is shown that the Shannon measure of uncertainty can be introduced into the Ackoff-Emery behavioural framework for dealing with measures of information. The greater clarity stemming from this partial unification of two apparently different approaches may help with the still formidable problems that are outlined.     

Possibilistic sequential decision making

When the information about uncertainty cannot be quantified in a simple, probabilistic way, the topic of possibilistic decision theory is often a natural one to consider. The development of possibilistic decision theory has lead to the proposition a series of possibilistic criteria, namely: optimistic and pessimistic possibilistic qualitative criteria [7], possibilistic likely dominance [2], [9], binary possibilistic utility [11] and possibilistic Choquet integrals [24]. This paper focuses on sequential decision making in possibilistic decision trees. It proposes a theoretical study on the complexity of the problem of finding an optimal strategy depending on the monotonicity property of the optimization criteria – when the criterion is transitive, this property indeed allows a polytime solving of the problem by Dynamic Programming. We show that most possibilistic decision criteria, but possibilistic Choquet integrals, satisfy monotonicity and that the corresponding optimization problems can be solved in polynomial time by Dynamic Programming. Concerning the possibilistic likely dominance criteria which is quasi-transitive but not fully transitive, we propose an extended version of Dynamic Programming which remains polynomial in the size of the decision tree. We also show that for the particular case of possibilistic Choquet integrals, the problem of finding an optimal strategy is NP-hard. It can be solved by a Branch and Bound algorithm. Experiments show that even not necessarily optimal, the strategies built by Dynamic Programming are generally very good.     

Fuzzy stochastic linear programming: Survey and future research directions

In many real-life problems one has to base decision on information which is both fuzzily imprecise and probabilistically uncertain. Although consistency indexes providing a union nexus between possibilistic and probabilistic representation of uncertainty exist, there are no reliable transformations between them. This calls for new paradigms for incorporating the two kinds of uncertainty into mathematical models. Fuzzy stochastic linear programming is an attempt to fulfill this need. It deals with modelling and problem solving issues related to situations where randomness and fuzziness co-occur in a linear programming framework. In this paper we provide a survey of the essential elements, methods and algorithms for this class of linear programming problems along with promising research directions. Being a survey, the paper includes many references to both give due credit to results in the field and to help readers obtain more detailed information on issues of interest.     

The weighted hook length formula

Based on the ideas in Ciocan-Fontanine, Konvalinka and Pak (2009) [5], we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. The first proof is completely bijective, and in a special case gives a new short combinatorial proof of the hook length formula. Our second proof is probabilistic, generalizing the (usual) hook walk proof of Greene, Nijenhuis and Wilf (1979) [15], as well as the q-walk of Kerov (1993) [20]. Further applications are also presented.     

The axiomatic power of Kolmogorov complexity

The famous Gödel incompleteness theorem states that for every consistent, recursive, and sufficiently rich formal theory T there exist true statements that are unprovable in T. Such statements would be natural candidates for being added as axioms, but how can we obtain them? One classical (and well studied) approach is to add to some theory T an axiom that claims the consistency of T  . In this paper we discuss another approach motivated by Chaitin's version of Gödel's theorem where axioms claiming the randomness (or incompressibility) of some strings are probabilistically added, and show that it is not really useful, in the sense that this does not help us prove new interesting theorems. This result answers a question recently asked by Lipton. The situation changes if we take into account the size of the proofs: randomly chosen axioms may help making proofs much shorter (unless NP=PSPACE$\mathrm{NP}=\mathrm{PSPACE}$).     

Strong guidance on mitigating the effects of uncertainties in the brass casting blending problem: a hybrid optimization approach

The scrap charge optimization problem in the brass casting process is a critical management concern that aims to reduce the charge while preventing specification violations. Uncertainties in scrap material compositions often cause violations in product standards. In this study, we have discussed the aleatory and epistemic uncertainties and modelled them by using probability and possibility distributions, respectively. Mathematical models including probabilistic and possibilistic parameters are generally solved by transforming one type of parameter into the other. However, the transformation processes have some handicaps such as knowledge losses or virtual information production. In this paper, we have proposed a new solution approach that needs no transformation process and so eliminates these handicaps. The proposed approach combines both chance-constrained stochastic programming and possibilistic programming. The solution of the numerical example has shown that the blending problem including probabilistic and possibilistic uncertainties can be successfully handled and solved by the proposed approach.     

On weak solutions of forward�Cbackward SDEs

In this paper we continue exploring the notion of weak solution of forward?Cbackward stochastic differential equations (FBSDEs) and associated forward?Cbackward martingale problems (FBMPs). The main purpose of this work is to remove the constraints on the martingale integrands in the uniqueness proofs in our previous work (Ma et?al. in Ann Probab 36(6):2092?C2125, 2008). We consider a general class of non-degenerate FBSDEs in which all the coefficients are assumed to be essentially only bounded and uniformly continuous, and the uniqueness is proved in the space of all the square integrable adapted solutions, the standard solution space in the FBSDE literature. A new notion of semi-strong solution is introduced to clarify the relations among different definitions of weak solution in the literature, and it is in fact instrumental in our uniqueness proof. As a by-product, we also establish some a priori estimates of the second derivatives of the solution to the decoupling quasilinear PDE.     

An extension to possibilistic fuzzy cluster analysis

We explore an approach to possibilistic fuzzy clustering that avoids a severe drawback of the conventional approach, namely that the objective function is truly minimized only if all cluster centers are identical. Our approach is based on the idea that this undesired property can be avoided if we introduce a mutual repulsion of the clusters, so that they are forced away from each other. We develop this approach for the possibilistic fuzzy c-means algorithm and the Gustafson–Kessel algorithm. In our experiments we found that in this way we can combine the partitioning property of the probabilistic fuzzy c-means algorithm with the advantages of a possibilistic approach w.r.t. the interpretation of the membership degrees.     

A mixed theory of information—X: Information functions and information measures

In the probabilistic theory of information, measures of information depend only upon the probabilities of the events, whereas in the nonprobabilistic theory these measures depend only upon the events. In a new, mixed theory of information the measures of information are assumed to depend on both the probabilities and the events. In this paper we consider measures depending upon a n-tuple of events and upon a finite number of n-ary complete, discrete probability distributions and characterize these measures of information only by two properties: by a recursivity condition, which states how the information changes by splitting one event of a system (one outcome of an experiment, market situation etc.) into two events, and by a weak symmetry condition (no regularity condition is assumed). Our result generalizes all recent results on this topic and especially we get from this one theorem a lot of characterization theorems for some well-known (purely probabilistic) information measures like the Shannon entropy, the entropy of degree α, the inaccuracy, the directed divergence, and the information improvement.     

Fuzzy coefficient volatility (FCV) models with applications

Recently, Carlsson and Fuller [C. Carlsson, R. Fuller, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems 122 (2001) 315–326] have introduced possibilistic mean, variance and covariance of fuzzy numbers and Fuller and Majlender [R. Fuller, P. Majlender, On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets and Systems 136 (2003) 363–374] have introduced the notion of crisp weighted possibilistic moments of fuzzy numbers. In this paper, we propose a class of FCV (Fuzzy Coefficient Volatility) models and study the moment properties. The method used here is very similar to the method used in Appadoo et al. [S.S. Appadoo, M. Ghahramani, A. Thavaneswaran, Moment properties of some time series models, Math. Sci. 30 (1) (2005) 50–63]. The proposed models incorporate fuzziness, subjectivity, arbitrariness and uncertainty observed in most financial time series. The usual forecasting method does not incorporate parameter variability. Fuzzy numbers are used to model the parameters to incorporate parameter variability.     

A jump-diffusion model for option pricing under fuzzy environments

Owing to fluctuations in the financial markets from time to time, the rate λ of Poisson process and jump sequence {V_i} in the Merton’s normal jump-diffusion model cannot be expected in a precise sense. Therefore, the fuzzy set theory proposed by Zadeh [Zadeh, L.A., 1965. Fuzzy sets. Inform. Control 8, 338-353] and the fuzzy random variable introduced by Kwakernaak [Kwakernaak, H., 1978. Fuzzy random variables I: Definitions and theorems. Inform. Sci. 15, 1-29] and Puri and Ralescu [Puri, M.L., Ralescu, D.A., 1986. Fuzzy random variables. J. Math. Anal. Appl. 114, 409-422] may be useful for modeling this kind of imprecise problem. In this paper, probability is applied to characterize the uncertainty as to whether jumps occur or not, and what the amplitudes are, while fuzziness is applied to characterize the uncertainty related to the exact number of jump times and the jump amplitudes, due to a lack of knowledge regarding financial markets. This paper presents a fuzzy normal jump-diffusion model for European option pricing, with uncertainty of both randomness and fuzziness in the jumps, which is a reasonable and a natural extension of the Merton [Merton, R.C., 1976. Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125-144] normal jump-diffusion model. Based on the crisp weighted possibilistic mean values of the fuzzy variables in fuzzy normal jump-diffusion model, we also obtain the crisp weighted possibilistic mean normal jump-diffusion model. Numerical analysis shows that the fuzzy normal jump-diffusion model and the crisp weighted possibilistic mean normal jump-diffusion model proposed in this paper are reasonable, and can be taken as reference pricing tools for financial investors.