Narrow Results

This paper defines the truncated normalized max product operation for the transformation ofstates of a network and provides a method for solving a set of equations based on this operation. The operation serves as the transformation for the set of fully connected units in a recurrent network that otherwise might consist of linear threshold units. Component values of the state vector and ouputs of the units take on the values in the set {0, 0.1, …, 0.9, 1}. The result is a much larger state space given a particular number of units and size of connection matrix than for a network based on threshold units. Since the operation defined here can form the basis of transformations in a recurrent network with a finite number of states, fixed points or cycles are possible and the network based on this operation for transformations can be used as an associative memory or pattern classifier with fixed points taking on the role of prototypes. Discrete fully recurrent networks have proven themselves to be very useful as associative memories and as classifiers. However they are often based on units that have binary states. The effect of this is that the data to be processed consisting of vectors in ℜⁿ have to be converted to vectors in {0, 1}^m with m much larger than n since binary encoding based on positional notation is not feasible. This implies a large increase in the number of components. The effect can be lessened by allowing more states for each unit in our network. The network proposed demonstrates those properties that are desirable in an associative memory very well as the simulations show.

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 this paper, a new kind of fuzzy relational equations (FREs for short) A ∘^R_*x = b is first introduced, and then the problem of solving solution to the FREs is discussed, where A is an m × n matrix, x and b are an n and an m dimensional column vectors, respectively. More specifically, their solvability and unique solvability are investigated, the corresponding necessary and sufficient conditions are presented, the complete solution set is obtained. It is worth noting the method to construct the complete solution set.

Given a Brouwerian complete lattice (L,≤) and two referential sets K and E, and using a fuzzy relation R ∈ L^E×E which is reflexive and symmetric, certain fuzzy relations are characterized as solutions of X ⊲ R = X, proving that they play the role of the blocks in the context of crisp tolerance relations. Moreover, it is verified that these new fuzzy blocks can be determinated by means of the L-fuzzy concepts associated with the K-labeled L-fuzzy context (L,K,E,E,R).

This paper considers the identification of problems which generate anomalies at firms through the observed symptoms on the basis of fuzzy relations and Zadeh's compositional rule of inference. A procedure for determining the fuzzy cause vector of an economic and financial diagnosis problem is proposed, which consists of the design of fuzzy relational matrix and the resolution of a system of fuzzy relational equations. An efficient algorithm for solving fuzzy relational equations in terms of the associated set covering problem is introduced. It utilizes a back-tracking method to generate each minimal covering, where no duplicate or non-minimal coverings exist. A numerical example of firms' insolvency causes diagnosis is also included.

An optimization-based methodology for on-line discrete-time fuzzy relational model identification is proposed. Approximated solutions for extended fuzzy relational equations are derived without the necessity of the previous identification of an appropriate initial fuzzy relation. A heuristic is used making the algorithm capable of solving the identification problem when the fuzzy relational matrix is biased. A set of quadratic performance indices containing only fuzzy quantities provides accurate results and allows the utilization of a simple optimization method. The efficiency of the proposed methodology is illustrated using two numerical examples in which synthetic data are generated by the identified models.