Information Granules Of Higher Type Research Articles

Computing and Clustering in the Environment of Order-2 Information Granules.

A visible trend in representing knowledge through information granules manifests in the developments of information granules of higher type and higher order, in particular, type-2 fuzzy sets and order-2 fuzzy sets. All these constructs are aimed at the formalization and processing data at a certain level of abstraction. Along the same line, in the recent years, we have seen intensive developments in fuzzy clustering, which are not surprising in light of a growing impact of clustering on fundamentals of fuzzy sets (as supporting ways to elicit membership functions) as well as algorithms (in which clustering and clusters form an integral functional component of various fuzzy models). In this study, we investigate order-2 information granules (fuzzy sets) by analyzing their formal description and properties to cope with structural and hierarchically organized concepts emerging from data. The design of order-2 information granules on a basis of available experimental evidence is discussed and a way of expressing similarity (resemblance) of two order-2 information granules by engaging semantically oriented distance is discussed. In the sequel, the study reported here delivers highly original contributions in the realm of order-2 clustering algorithms. Formally, the clustering problem under discussion is posed as follows: given is a finite collection of reference information granules. Determine a structure in data defined over the space of such granules. Conceptually, this makes a radical shift in comparison with data defined in the p -dimensional space of real numbers Rp. In this situation, expressing distance between two data deserves prudent treatment so that such distance properly captures the semantics and consequently, the closeness between any two information granules to be determined in cluster formation. Following the proposal of the semantically guided distance (and its ensuing design process), we develop an order-2 variant of the fuzzy C-means (FCM), discuss its detailed algorithmic steps, and deliver interpretation of the obtained clustering results. Several relevant applied scenarios of order-2 FCM are identified for spatially and temporally distributed data, which deliver interesting motivating arguments and underline the practical relevance of this category of clustering. Experimental studies are provided to further elicit the performance of the clustering method and discuss essential ways of interpreting results.

A Development of Hierarchically Structured Granular Models Realized Through Allocation of Information Granularity

In this article, we elaborate on a design methodology and the detailed realization of hierarchically structured granular models by engaging the fundamental principles and concepts of granular computing. The existing models are elevated to a more abstract (general) level by allocating a certain level of information granularity throughout the parameter space. In a concise way, the essence of the overall architecture of the proposed modeling mechanism could be generalized as follows: Numeric model (granular model of type-0)→granular model of type-1→granular model of type-2→…→granular model of higher type. The results of the granular models come in the form of type-0, type-1, or higher type information granules, which are decided by the overall level of hierarchy of the corresponding granular model. The specificity of granular outputs becomes a more comprehensive and sound quantification of the prediction accuracy and precision of the model and the quality of the specific prediction outputs. The proposed method facilitates effective communication with humans, who could get actively involved in the modeling process and determine the suitable level of abstraction depending upon the requirements of the problem. The determination of a suitable level of information granularity is realized with the guidance of the principle of justifiable granularity. A number of experimental studies concerning publicly available datasets are presented to illustrate the development methodology and show the effectiveness of the approach to form hierarchically structured solutions (reflecting different levels of abstraction) to the problem.

Development of information granules of higher type and their applications to granular models of time series

The study is devoted to the design of information granules of higher type (especially type-2) with the use of the principle of justifiable granularity. The development of granules is realized in two key phases: first, information granules of type-1 are formed and then they are extended to type-2 constructs. Following the principle, information granules are designed by establishing a sound balance between their experimental justification (legitimacy) and specificity (associated with their underlying semantics). The definitions of coverage and specificity of type-2 information granules are revised to capture the essence of these constructs. Detailed formulas are derived for several main categories of membership functions (namely, triangular, parabolic, and square root) as well as intervals. The study delivers detailed results for interval-valued fuzzy sets described by membership functions coming from the main classes listed above. Illustrative studies include synthetic data exhibiting some probabilistic properties. The direct application of information granules of type-1 and type-2 is demonstrated in the description and prediction of time series realized in the setting of information granules (with the resulting models referred to as granular models of time series).

Granular Encoders and Decoders: A Study in Processing Information Granules

Information granules are generic building blocks supporting the processing realized in granular computing and facilitating communication with the environment. In this paper, we are concerned with a fundamental problem of encoding–decoding of information granules. The essence of the problem is outlined as follows: given a finite collection of granular data X 1, X 2,…, XN (sets, fuzzy sets, etc.), construct an optimal codebook composed of information granules A 1 , A2 , …, Ac , where typically c N, so that any Xk represented in terms of A i 's and then decoded (reconstructed) with the help of this codebook leads to the lowest decoding error. A fundamental result is established, which states that in the proposed encoders and decoders, when encoding–decoding error is present, the information granule coming as a result of decoding is of a higher type than the original information granules (say, if Xk is information granule of type-1, then its decoded version becomes information granule of type-2). It would be beneficial to note that as the encoding–decoding process is not lossless (in general, with an exception of a few special cases), the lossy nature of the method is emphasized by the emergence of information granules of higher type (in comparison with the original data being processed). For instance, when realizing encoding–decoding of numeric data (viz., information granules of type-0), the losses occur and they are quantified in terms of intervals, fuzzy sets, probabilities, rough sets, etc., where, in fact, the result becomes an information granule of type-1. In light of the nature of the constructed result when Xk is an interval or a fuzzy set, an optimized performance index engages a distance between the bounds of the interval-valued membership function. We develop decoding and encoding mechanisms by engaging the theory of possibility and fuzzy relational calculus and show that the decoded information granule is either a granular interval or interval-valued fuzzy set. The optimization mechanism is realized with the aid of the particle swarm optimization (PSO). A series of experiments are reported with intent to illustrate the details of the encoding–decoding mechanisms and show that the PSO algorithm can efficiently optimize the granular codebook.

Algorithmic Developments of Information Granules of Higher Type and Higher Order and Their Applications

Algorithmic Developments of Information Granules of Higher Type and Higher Order and Their Applications

Hierarchical Granular Clustering: An Emergence of Information Granules of Higher Type and Higher Order

In this study, we introduce a concept of hierarchical granular clustering and establish its algorithmic framework. We show that the proposed model naturally gives rise to information granules that are both of higher order and higher type, offering a compelling justification behind their emergence. In a concise way, we can capture the overall architecture of information granules as a hierarchy exhibiting conceptual layers of increasing abstraction: numeric data → information granules → information granules of type-2, order-2 → … information granules of higher type/order. The elevated type of information granules is reflective of the visible hierarchical facet of processing and the inherent diversity of the individual locally revealed structures in data. While the concept and the methodology deliver some general settings, the detailed algorithmic aspects are discussed in detail when using fuzzy clustering realized by means of fuzzy c-means. Furthermore, for illustrative purposes, we mainly focus on interval-valued fuzzy sets and granular interval fuzzy sets arising at the higher level of the hierarchy. Higher type fuzzy sets are formed with the help of the principle of justifiable granularity. The conceptually sound hierarchy is established in a general way, which makes it equally applicable to various formalisms of representation of information granules. Experiments are reported for synthetic and publicly available datasets.

Clustering Granular Data and Their Characterization With Information Granules of Higher Type

The study is devoted to the clustering of granular data and an evaluation of the results of such clustering. A comprehensive and systematic approach is developed, which is composed of three fundamental phases: 1) representation of granular data; 2) clustering carried out in the representation space of information granules; and 3) evaluation of quality of clusters following the reconstruction criterion. The reconstruction criterion formed originally for numeric data and leading to an idea of granular prototypes is revisited. We show here an emergence of granular information of higher type, which are used to implement granular interval prototypes. We discuss a way of forming granular data in the context of representation of time series and present clustering of granular time series.

Granular Computing and Human-Centricity in Computational Intelligence

Information granules and ensuing Granular Computing offer interesting opportunities to endow processing with an important facet of human-centricity. This facet implies that the underlying processing supports non-numeric data inherently associated with the variable perception of humans. Systems that commonly become distributed and hierarchical, managing granular information in hierarchical and distributed architectures, is of growing interest, especially when invoking mechanisms of knowledge generation and knowledge sharing. The outstanding feature of human centricity of Granular Computing along with essential fuzzy set-based constructs constitutes the crux of this study. The author elaborates on some new directions of knowledge elicitation and quantification realized in the setting of fuzzy sets. With this regard, the paper concentrates on knowledge-based clustering. It is also emphasized that collaboration and reconciliation of locally available knowledge give rise to the concept of higher type information granules. Other interesting directions enhancing human centricity of computing with fuzzy sets deals with non-numeric semi-qualitative characterization of information granules, as well as inherent evolving capabilities of associated human-centric systems. The author discusses a suite of algorithms facilitating a qualitative assessment of fuzzy sets, formulates a series of associated optimization tasks guided by well-formulated performance indexes, and discusses the underlying essence of resulting solutions.

Hierarchical Architectures of Fuzzy Models: From Type-1 fuzzy sets to Information Granules of Higher Type

Complex phenomena are perceived from different perspectives, diversified conceptual points of view and at various levels of granularity. Symbolic and sub-symbolic processing becomes an inherently visible computing practice. Distributed nature of perception becomes reflected in topologies of multi-agent systems. All of these facets challenge the well-established paradigms of system modeling including fuzzy models and neural networks. In spite of the diversity of existing architectures and underlying algorithms, a vast majority of fuzzy models adheres to the surprisingly homogeneous principles of Granular Computing, that are associated with the processing of granular information. In this study, being cognizant of this underpinning, we concentrate on the architectures and fundamentals supporting the reconciliation and characterization of a family of fuzzy models aimed at the representation of the same system (phenomenon) from different cognitive perspectives. The variety of points of view is reflected in dif...

Hierarchical Architectures of Fuzzy Models: From Type-1 fuzzy sets to Information Granules of Higher Type

Hierarchical Architectures of Fuzzy Models: From Type-1 fuzzy sets to Information Granules of Higher Type