Granular Computing

This article commences with a brief historical introduction to interval analysis and some applications in engineering. This is followed by a simple example motivating the use of interval analysis. A more detailed definition of interval analysis and some properties are then given followed by a discussion of the application of interval analysis to the problem of computing inclusions for the range of functions. A brief discussion of new types of interval based algorithms (mainly the interval Newton algorithm) is provided. The article is completed with a discussion of some application areas for interval analysis without claiming to be a complete survey.

In this Chapter, we present results derived in the context of state estimation of a class of real-life systems that are driven by some poorly known factors. For these systems, the representation of uncertainty as confidence intervals or the ellipsoids offers significant advantages over the more traditional approaches with probabilistic representation of noise. While the filtered-white-Gaussian noise model can be defined on grounds of mathematical convenience, its use is necessarily coupled with a hope that an estimator with good properties in idealised noise will still perform well in real noise. With good knowledge of the plant and its environment, a sufficiently accurate approximation to the probability density function can be obtained, but shortage of prior information or excessive computing demands normally rule out this option. A more realistic approach is to match the noise representation to the extent of prior knowledge. The relative merits of interval and ellipsoidal representations of noise are discussed in a set theoretic setting and are illustrated using both a simple synthetic example and a real-life scenario of state estimation of a water distribution system.

This paper deals with the estimation of the parameters of a model from experimental data. The aim of the method presented is to characterize the set S of all values of the parameter vector that are acceptable in the sense that all errors between the experimental data and the corresponding model outputs lie between known lower and upper bounds. This corresponds to what is known as bounded-error estimation or membership-set estimation. Most of the methods available to give guaranteed estimates of S rely on the hypothesis that the model output is linear in its parameters, contrary to the method advocated here which can deal with nonlinear models. This is made possible by the use of the tools of interval analysis, combined with a branch-and-bound algorithm. The purpose of the present paper is to show that the approach can be cast into the more general framework of granular computing.

The relevance, applicability and importance of fuzzy sets is generally linked to successful applications in the domain of engineering, especially when subjective notions are modelled and matched with data. For problems in which uncertainty has been modelled using probability theory in the past, discussions on what approach is right, frequently conclude that both should complement each other. In the present text, we consider such synergy of fuzzy sets, probability and possibility distributions provided by the concept of a random-set. Following a brief review of basic mathematical and semantic aspects of random-set theory, we introduce an application of the theory to time series analysis.

In recent years we witness a rapid growth of interest in rough set theory and its applications, worldwide. The theory has been followed by the development of several software systems that implement rough set operations, in particular for solving knowledge discovery and data mining tasks. Rough sets are applied in domains, such as, for instance, medicine, finance, telecommunication, vibration analysis, conflict resolution, intelligent agents, pattern recognition, control theory, signal analysis, process industry, marketing, etc.

We introduce basic notions and discuss methodologies for analyzing data and surveys some applications. In particular we present applications of rough set methods for feature selection, feature extraction, discovery of patterns and their applications for decomposition of large data tables as well as the relationship of rough sets with association rules. Boolean reasoning is crucial for all the discussed methods.

We also present an overview of some extensions of the classical rough set approach. Among them is rough mereology developed as a tool for synthesis of objects satisfying a given specification in a satisfactory degree. Applications of rough mereology in such areas like granular computing, spatial reasoning and data mining in distributed environment are outlined.

Defuzzification is an important operation in the theory of fuzzy sets. It transforms a fuzzy set information into a numeric data information. This operation along with the operation of fuzzification is critical to the design of fuzzy systems as both of these operations provide nexus between the fuzzy set domain and the real valued scalar domain. We need the synergy of both of these domains to solve many of our ill-posed problems effectively. In this paper, we will address the problem of defuzzification, present merits and demerits of various defuzzification strategies that are used in the theory and practice, and in design and implementation of applications involving fuzzy theory, fuzzy control, and fuzzy rule base, and fuzzy inference-based systems. We also present in this paper a simple and yet novel defuzzification mechanism.

In this chapter, we propose two new algorithms to infer automatically a fuzzy partition for the universe of a set of values, when each of these values is associated with a class. These algorithms are based on the use of mathematical morphology operators that are used to filter the given set of values and highlight kernel for fuzzy subsets. Their purpose is to be used in an inductive learning algorithm to construct fuzzy partitions on numerical universes of values.

We present a coding method for linguistic variables which we have named Incremental Discretization. It allows us to express any fuzzy subset of the universe of discourse of the linguistic variable in binary or bipolar terms. This will permit us to process fuzzy information expressed in linguistic terms using discrete models of Artificial Intelligence. In order to test the effectiveness, we apply this method to the memorization of a set of fuzzy rules using models of discrete associative memories.

Fuzzy quantification is a linguistic granulation technique capable of expressing the global characteristics of a collection of individuals, or a relation between individuals, through meaningful linguistic summaries. However, existing approaches to fuzzy quantification fail to provide convincing results in the important case of two-place quantification (e.g. “many blondes are tall”). We develop an axiomatic framework for fuzzy quantification which complies with a large number of linguistically motivated adequacy criteria. In particular, we present the first models of fuzzy quantification which provide an adequate account of the “hard” cases of multiplace quantifiers, non-monotonic quantifiers, and non-quantitative quantifiers, and we show how the resulting operators can be efficiently implemented based on histogram computations.¹

The structure of fuzzy models produced by a heursitic analysis of the problem domain is compared with that of models algorithmically generated from training data. The trade-offs between granularity, specificity, interpretability, and efficiency are examined for rule-bases produced in each of these manners. An algorithm that combines rule learning with region merging is introduced to incorporate beneficial features of both the heuristic and learning approaches to producing fuzzy models.

The basic premise of granular computing is that, by reducing precision in our model of a system, we can suppress minor details and focus on the most significant relationships in the system. In this chapter, we will test this premise by defining a granular neural network and testing it on the Iris data set. Our hypothesis is that the granular neural network will be able to learn the Iris data set, but not as accurately as a standard neural network. Our network is a novel neuro-fuzzy systems architecture called the linguistic neural network. The defining characteristic of this network is that all connection weights are linguistic variables, whose values are updated by adding linguistic hedges. We define two new hedges, whose semantics require a generalization of the standard definition of linguistic variables. These generalized linguistic variables lead naturally to a linguistic arithmetic, which we prove forms a vector space. The node functions of the linguistic neural network are defined in terms of this linguistic arithmetic. The learning method used for the network is a modified Backpropagation algorithm, with the original arithmetic operations replaced by their linguistic equivalents. In a simulation experiment, this granulated version of the multilayer perceptron achieved 90% accuracy on the Iris data set, using a coarse granulation. This result supports our hypothesis.

The goal of input-output modeling is to apply a test input to a system, analyze the results, and learn something useful from the cause-effect pair. Any automated modeling tool that takes this approach must be able to reason effectively about sensors and actuators and their interactions with the target system. The granulation level of the information involved in this process ranges from low-level data analysis techniques to abstract, qualitative observations about the system. This chapter describes a knowledge representation and reasoning framework that allows this process to be automated.

The paper deals with optical music recognition (OMR) as a process of structured data processing applied to music notation. Granularity of OMR in both its aspects: data representation and data processing is especially emphasised in the paper. OMR is a challenge in intelligent computing technologies, especially in such fields as pattern recognition and knowledge representation and processing. Music notation is a language allowing for communication in music, one of most sophisticated field of human activity, and has a high level of complexity itself. On the one hand, music notation symbols vary in size and have complex shapes; they often touch and overlap each other. This feature makes the recognition of music symbols a very difficult and complicated task. On the other hand, music notation is a two dimensional language in which importance of geometrical and logical relations between its symbols may be compared to the importance of the symbols alone. Due to complexity of music nature and music notation, music representation, necessary to store and reuse recognised information, is also the key issue in music notation recognition and music processing. Both: the data representation and the data processing used in OMR is highly structured, granular rather than numeric. OMR technology fits paradigm of granular computing

In this chapter, we present a new approach for MPEG variable bit rate (VBR) video modeling using a type-2 fuzzy logic system (FLS). We demonstrate that a type-2 fuzzy membership function, i.e., a Gaussian MF with uncertain variance, is most appropriate to model the log-value of I/P/B frame sizes in MPEG VBR video. We treat the video traffic as a dynamic system, and use a type-2 FLS to model this system. Simulation results show that a type-2 FLS performs much better than a type-1 FLS in video traffic modeling.

One of the most important problems on rule induction methods is that they cannot extract the rules that plausibly represent experts’ decision processes: the induced rules are too short to represent the reasoning of domain experts. In this paper, the characteristics of experts’ rules are closely examined and a new approach to extract plausible rules is introduced, which consists of the following three procedures. First, the characterization of decision attributes (given classes) is extracted from databases and the classes are classified into several groups with respect to the characterization. Then, two kinds of sub-rules, characterization rules for each group and discrimination rules for each class in the group are induced. Finally, those two parts are integrated into one rule for each decision attribute. The proposed method was evaluated on medical databases, the experimental results of which show that induced rules correctly represent experts’ decision processes.