Granular Models: Design Insights and Development Practices

Granular Models: Design Insights and Development Practices

Witold Pedrycz (Polish Academy of Sciences, Poland) and Athanasios Vasilakos (Polish Academy of Sciences, Poland)
DOI: 10.4018/978-1-60566-324-1.ch010

Abstract

In contrast to numeric models, granular models produce results coming in a form of some information granules. Owing to the granularity of information these constructs dwell upon, such models become highly transparent and interpretable as well as operationally effective. Given also the fact that information granules come with a clearly defined semantics, granular models are often referred to as linguistic models. The crux of the design of the linguistic models studied in this paper exhibits two important features. First, the model is constructed on a basis of information granules which are assembled in the form of a web of associations between the granules formed in the output and input spaces. Given the semantics of information granules, we envision that a blueprint of the granular model can be formed effortlessly and with a very limited computing overhead. Second, the interpretability of the model is retained as the entire construct dwells on the conceptual entities of a well-defined semantics. The granulation of available data is accomplished by a carefully designed mechanism of fuzzy clustering which takes into consideration specific problem-driven requirements expressed by the designer at the time of the conceptualization of the model. We elaborate on a so-called context – based (conditional) Fuzzy C-Means (cond-FCM, for brief) to demonstrate how the fuzzy clustering is engaged in the design process. The clusters formed in the input space become induced (implied) by the context fuzzy sets predefined in the output space. The context fuzzy sets are defined in advance by the designer of the model so this design facet provides an active way of forming the model and in this manner becomes instrumental in the determination of a perspective at which a certain phenomenon is to be captured and modeled. This stands in a sharp contrast with most modeling approaches where the development is somewhat passive by being predominantly based on the existing data. The linkages between the fuzzy clusters induced by the given context fuzzy set in the output space are combined by forming a blueprint of the overall granular model. The membership functions of the context fuzzy sets are used as granular weights (connections) of the output processing unit (linear neuron) which subsequently lead to the granular output of the model thus identifying a feasible region of possible output values for the given input. While the above design is quite generic addressing a way in which information granules are assembled in the form of the model, we discuss further refinements which include (a) optimization of the context fuzzy sets, (b) inclusion of bias in the linear neuron at the output layer.
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2. The Cluster-Based Representation Of The Input–Output Mappings

Clusters [2] and fuzzy clusters [1][3][4][6][10][19][20] establish a sound basis for constructing fuzzy models [17][18]. By forming fuzzy clusters in the input and output spaces (spaces of input and output variables), we span the fuzzy model over a collection of prototypes. More descriptively, these prototypes are regarded as a structural skeleton or a design blueprint of the resulting model. Once the prototypes have been formed, there are several ways of developing the detailed expressions governing the detailed relationships of the model. The one commonly encountered in the literature takes the prototypes formed in the output space, that is z1, z2, …,zcR and combines them linearly by using the membership grades of the corresponding degrees of membership of the fuzzy clusters in the input space. Consider some given input x. Denote the corresponding grades of membership produced by the prototypes v1, v2, …, and vc located in the input space by u1(x), u2(x),…, uc(x). The output of the model reads as follows

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