Eigenvalues and eigenvectors are widely used in various applications. Particularly, these concepts underlie analysis of consistency of a decision maker’s (DMs) preference knowledge. In real-world problems, DMs knowledge is inherently associated with imprecision and partial reliability. This involves combination of fuzzy and probabilistic information. The concept of a Z-number is a formal construct to describe such kind of information. In this study, we formulate the concepts of Z-number valued eigenvalue and eigenvector for matrices components of which are Z-numbers. A formal statement of the problem and a solution method for computation of Z-number valued eigensolutions are proposed. Numerical examples and an application devoted to foreign market selection problem are provided to show the usefulness of the proposed approach.
In this study, we introduce a new category of fuzzy inference systems based on data (information) granulation and show their applications to the identification of complex and usually nonlinear systems. Information granules are treated as collections of objects (data, in particular) brought together by the criteria of proximity, similarity, or functionality. The formal framework of information granulation along with the information granules themselves become an important design feature of fuzzy models, which in essence are geared towards capturing relationship between information granules rather than plain numeric data. The key characteristics of experimental data being used in the construction of the fuzzy model are carefully reflected by fuzzy rules formed therein. Information granulation realized with the aid of Hard C-Means (HCM) clustering helps determine the initial values of the parameters of the fuzzy models. This in particular concerns such important components of the rules as the initial apexes of the membership functions standing in the premise part of the fuzzy rules and the initial values of the polynomial functions present in their consequence part. The initial values of the parameters are tuned effectively with the aid of the genetic algorithms (GAs) and the least square method (LSM). An aggregate objective function is constructed in order to strike a sound balance between the approximation and generalization capabilities of the fuzzy model. The model is evaluated with the use of numerical experimentation and contrasted with the quality of some conventional fuzzy models already encountered in the literature.
The paper elaborates on the encoding and decoding of numerical and nonnumerical data. Proposed are general criteria leading to the distortion-free interfacing mechanisms that help transform information between the systems (or modelling environments) operating at different levels of information granularity. Distinguished are three basic categories of information: numerical, interval-valued, and linguistic (fuzzy). As all of them are dealt with here, the paper subsumes the current studies concentrated exclusively on representing fuzzy sets through their numerical representatives (prototypes). The algorithmic framework in which the distortion-free interfacing is completed is realized through neural networks. Each category of information is treated separately and gives rise to its own specialized architecture of the neural network. Similarly, these networks require carefully designed training sets that fully capture the specificity of the reconstruction problem. Several carefully selected numerical examples are aimed at the illustration of the key ideas.
Accurate classification of biomedical spectra is often difficult due to the large number of features, which tends to have a confounding effect. We present a strategy where the original spectral feature space is transformed using a fuzzy set theoretic method, which analyzes the features' interquartile ranges, coupled with a stochastic feature selection mechanism, which identifies highly discriminatory feature subsets. We demonstrate the effectiveness of this strategy using biofluid data acquired from a magnetic resonance spectrometer
Introduces a notion of relevance (conceptual stability) of information granules. Granulation of data results in a series of chunks of information usually referred to as information granules. These information granules are basic building entities involved in the design of a broad class of systems. Information granules are also percepts-entities being perceived by humans as being essential while working with some real-world phenomena, especially describing and interacting with them. The percepts need to be comprehensible. They should also reflect the experimental evidence. Furthermore, information granules should be stable, meaning that they reconcile experimental reality with the subjective and ultimately observer-based judgement about the environment. Once being stable, information granules could be viewed as independent. The proposed environment supporting this concept dwells on the ideas of statistical inference that helps quantify stability thorough nonparametric testing.
