International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems

This paper shows how Fuzzy Set Theory can be used in investment analysis when, as usual, these investments are developed under uncertainty, i.e. the investor has only subjective estimates based on his experience or knowledge about the future cash-flows of the investments, the discount rate, etc. In particular, we will develop basic concepts for investment analysis as the Net Present Value and the Internal Rate of Return by assuming that the initial data are fuzzy numbers. Later we will analyse how to rank investments and how to select the tangible investment portfolios when the magnitudes are estimated subjectively by comparing fuzzy numbers and with possibilistic mathematical programming.

Shannon's entropy and Simpson's index are the most used measures of species diversity. As the Simpson index proves to be just an approximation of the Shannon entropy, conditional Simpson indices of diversity and a global measure of interdependence among species are introduced, similar to those used in the corresponding entropic formalism from information theory. Also, since both the Shannon entropy and the Simpson index depend only on the number and relative abundance of the respective species in a given ecosystem, the paper generalizes these indices of diversity to the case when a numerical weight is attached to each species. Such a weight could reflect supplementary information about the absolute abundance, the economic significance, or the conservation value of the species.

A hierarchical uncertainty model for combining different evidence is studied in the paper. The model is general enough for many applications. The presented approach for dealing with the model allows us to combine the available heterogeneous information in the following ways: computing new probability bounds for some predefined interval of previsions, computing an "average" interval of first-order previsions, and updating the second-order probabilities after observing new events. Numerical examples illustrate this approach.

Two fundamental questions on the extensions of many-valued equivalence relations, whose solutions have a crucial concern for the applications of many-valued equivalence relations, are explicitly stated to characterize the extensions of many-valued equivalence relations. Thereafter introducing the satisfactory solutions of these questions, the desirable characterizations of the extensions of many-valued equivalence relations are pointed out.

A new approach to normalizing fuzzy sets is introduced where it is assumed that the normalization method is compatible with a given t-norm. In this context it is proved that the most usual ways to normalize fuzzy subsets correspond to the most common t-norms.

For a given fuzzy subset μ, the corresponding normalized fuzzy subset can be viewed as the distribution of μ conditioned on the (degree of) existence of its elements with maximal membership. From this view point we investigate the less specific normal fuzzy subset of X among the most similar fuzzy subsets to μ and the normal fuzzy subset generating the same fuzzy T-preorder as μ.

An iterative method of solving a set of equations based on the truncated normalized max product is described. The operation may serve as the transformation for the set of fully connected units in a fully recurrent network that might otherwise consist of linear threshold units. Because of truncation and normalization the network acting under this transformation has a finite number of states and components of the state vector are bounded. Component values however are not restricted to binary values as would be the case if the network consisted of linear threshold units but can now take on the values in the set {0, 0.1,..0.9, 1}. This means that each unit although still having discrete output can provide finer granularity compared to the case where a linear threshold unit is used. Truncation is natural in hardware implementation where only a finite number of places behind the decimal are retained.

In recent years there has been a spate of papers describing systems for probabilistic reasoning which do not use numerical probabilities. In some cases these systems are unable to make any useful inferences because they deal with changes in probability at too high a level of abstraction. This paper discusses one of the problems this level of abstraction can cause, and shows how the use of a technique for order of magnitude reasoning can reduce its impact.

This section is maintained by Vladik Kreinovich. Please send you abstract (or copies of papers that you want to see reviewed here) to vladik@cs.utep.edu, or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.