Pattern classification problems and fuzzy sets

A unified presentation of classical clustering algorithms is proposed both for the hard and fuzzy pattern classification problems. Based on two types of objective functions, a new method is presented and compared with the procedures of Dunn and Ruspini. In order to determine the best, or more natural number of fuzzy clusters, two coefficients that measure the “degree of non-fuzziness” of the partition are proposed. Numerous computational results are shown.     

The fuzzy entropy concept and its application

The fuzzy entropy is applied to the seal impression problem to measure the subjective value of information under the condition of uncertainty.Two methods, successive and direct segmentation, are introduced in recognizing the pattern to find out which method is more cost-effective. For the equal entropy, the comparative elements sought in association with the cost information on the successive and direct segmentation methods are 441 and 1024, respectively. Thus, the effectiveness of the former method is 2.32 times higher than that of the latter method, provided that the cost of information is equal.A parametric analysis among the membership function, fuzzy entropy, and threshold level is made and a summary graph shown.     

Some comments on fuzzy variables

Nahmias introduced the concept of a fuzzy variable as a possible axiomatic framework from which a rigorous theory of fuzziness may be constructed. In this paper we attempt to shed more light on fuzzy variables in analogy with random variables. In particular, we study the problem: if X₁, X₂,…,X_n are mutually unrelated fuzzy variables with common membership function μ and α₁, α₂,…,α_n are real numbers satisfying α_i ? o for every i and Σ_i=1ⁿα_i=1, when does does Z = Σ_{i = 1}ⁿα_iX_i have the same membership function μ?     

Optimization schemes for decomposition of fuzzy relations

The paper proposes an optimization environment for decomposing fuzzy relations. Two basic categories of approaches are developed: (i) based on direct gradient-oriented optimization methods, and (ii) exploiting the ideas of fuzzy neurocomputation. These two approaches are carefully analyzed and contrasted making use of illustrative numerical material. Some immediate applications to pattern classification and fuzzy interpolation are studied in detail. The study includes also some generalizations of the generic decomposition problem.     

Pattern recognition based Kalman filter for indoor localization using TDOA algorithm

In this paper, we will present a motion pattern recognition based Kalman filter (PRKF), and apply it to the time difference of arrival (TDOA) algorithm of indoor localization. The state matrix in Kalman filter (KF) is determined by the motion pattern which the target node is supposed to act, and this will bring new system error if the assumption is not correct. Considering this, we first create three fuzzy sets using three KFs whose state matrix stand for different motion patterns, then linearly combined the memberships of a target node of the fuzzy sets. Finally, simulation results show that the PRKF can enhance the localization accuracy about more than 20%.     

Energy discriminant analysis, quantum logic, and fuzzy sets

In this paper, we show that quantum logic of linear subspaces can be used for recognition of random signals by a Bayesian energy discriminant classifier. The energy distribution on linear subspaces is described by the correlation matrix of the probability distribution. We show that the correlation matrix corresponds to von Neumann density matrix in quantum theory. We suggest the interpretation of quantum logic as a fuzzy logic of fuzzy sets. The use of quantum logic for recognition is based on the fact that the probability distribution of each class lies approximately in a lower-dimensional subspace of feature space. We offer the interpretation of discriminant functions as membership functions of fuzzy sets. Also, we offer the quality functional for optimal choice of discriminant functions for recognition from some class of discriminant functions.     

Edge search in graphs with restricted test sets

Suppose a graph G(V,E) contains one defective edge e. We search for the endpoints of e by asking questions of the form “Is at least one of the vertices of X an endpoint of e?”, where X is a subset of V with cardinality at most p. Then what is the minimum number c_p(G) of questions, which are needed in the worst case to find e?We solve this search problem suggested by M. Aigner in [M. Aigner, Combinatorial Search, Teubner, 1988] by deriving lower and sharp upper bounds for c_p(G). For the case that G is the complete graph K_n the problem described above is equivalent to the (2,n) group testing problem with test sets of cardinality at most p. We present sharp upper and lower bounds for the worst case number c_p of tests for this group testing problem and show that the maximum difference between the upper and the lower bounds is 3.     

Analysis of maximum total return in the continuous knapsack problem with fuzzy object weights

This paper proposes a parametric programming approach to analyze the fuzzy maximum total return in the continuous knapsack problem with fuzzy objective weights, in that the membership function of the maximum total return is constructed. The idea is based on Zadeh’s extension principle, α-cut representation, and the duality theorem of linear programming. A pair of linear programs parameterized by possibility level α is formulated to calculate the lower and upper bounds of the fuzzy maximum total return at α, through which the membership function of the maximum total return is constructed. To demonstrate the validity of the proposed procedure, an example studied by the previous studies is investigated successfully. Since the fuzzy maximum total return is completely expressed by a membership function rather than by a crisp value reported in previous studies, the fuzziness of object weights is conserved completely, and more information is provided for making decisions in real-world resource allocation applications. The generalization of the proposed approach for other types of knapsack problems is also straightforward.     

Applications of fuzzy sets in curve fitting

This paper presents several methods for functional approximation with variable-knot variable-degree splines, with variable-knot first order splines, which are relatively easy to find, as the intermediate input. By means of a fuzzy characteristic function determining how “sharp” the angle at each knot is (the higher the polynomial degree, the “sharper” its first order spline approximation bends), we can decide whether we can group certain adjacent segments together to be approximated by a single higher order polynomial segment. Some simulation experiments have also been done.     

Integration of fuzzy linguistic information in language comprehension

To successfully understand a natural language utterance, a person must simultaneously consider information about its perceptual, syntactic, semantic, and pragmatic structure. At each of these levels of analysis, the linguistic information is continuously but systematically variable. It is argued that in order to cope with and, in fact, exploit this continuous information requires fuzzy propositional representations of linguistic knowledge. Such a fuzzy propositional theoretical framework is proposed and models of several specific language processing components are developed and empirically evaluated within this framework. The model system incorporates many of the major principles of current cognitive psychological theory and extends them into the fuzzy domain.