Research On FCM-based Granulation-Degranulation Mechanisms

As an important technology in Artificial Intelligence(AI)Granular Computing(Gr C)has emerged as a new multi-disciplinary paradigm and received much attention in recent years.Information granules have been considered to be the fundamental constructs of Granular Computing.As a useful unsupervised learning technique,clustering has become a powerful approach to build information granules.It offers a suite of algorithms aiming at the discovery of a structure in the given dataset.This type of approach partitions a given input space into several regions,depending upon some preselected similarity measures.One of the most widely used and effective clustering approaches is Fuzzy C-Means(FCM).From the general viewpoint,the FCM is usually regarded as a granular information technique,where information granule is represented(encoding,with the aid of constructed prototypes and partitions,data are encoded into information granules.In other words,cluster prototypes and partition matrices are obtained by optimizing the fuzzy set-based clustering model)by its prototype(center of the cluster)and a partition matrix(both descriptors are numeric).Clustering granular instead of numeric data provides a novel and interesting avenue of investigation.Numeric data are described as prototypes and partition matrices,which is the so-called granulation mechanism.In the FCM-based granulation progress,fuzzy clustering approaches are used to cluster the numerical data into fuzzy information granules.Degranulation,as an inverse problem of granulation that involves the reconstruction of numeric results on the basis of already constructed information granules,is also task worth studying.The reconstruction is usually referred to as a degranulation or decoding process.To some extent,the degranulation can also reflect the performance of granulation mechanism(classification performance of the fuzzy clustering).The mechanism of degranulation also involves a series of processes of dealing with fuzzy information granules.It plays an important role in Gr C,just as analogto-digital(A/D)conversion as well as digital-to-analog(D/A)conversion in the field of signal processing,and fuzzification-defuzzification in the field of fuzzy control systems.The classification rate and reconstruction(degranulation)error are often used as the performance evaluation indexes of the granulation-degranulation mechanism.Previous studies indicate that the reconstruction(degranulation)and the classification(granulation)are related to each other.So far,the topic of granulation-degranulation mechanism has not been intensively studied in the world.The lack of a well-established body of knowledge opens up new opportunities and challenges but also calls for more investigations in this area.The main research in this paper is the FCM-based granulation-degranulation mechanism,that is,the construction and decomposition of the information granules,which are presented as follows:In this research,the location of the prototypes are stacked as a matrix,then a novel matrix transformation model of granulation-degranulation mechanism is established.The new model of granulation-degranulation mechanism describes the transformation relationship between the numerical data subspace and prototype subspace,and the mathematical essence of the granulation-degranulation mechanism is revealed.The the following studies are carried out on the basis of the new model.Based on the subspace transformation model of the granulation-degranulation mechanism,firstly two enhanced schemes of granulation mechanism(Bi-fuzzy granulation mechanism,Data weighted granulation mechanism)are designed.Under the assumption that the information granules can be ideally decomposed(degranulation error is zero)and supervised by the degranulation mechanism,a rotation invariant structure of prototype subspace is constructed.Then a bi-fuzzy granulation scheme is obtained through a series of mathematical analysis.In the data weighted granulation mechanism,a dataset is divided into boundary data and non-boundary data.The partition matrix is used to determine the boundary data and the non-boundary data.Then,different weights are set to each datum to construct the weighted data.During this process,the weights for the boundary data and the non-boundary data are quite different,which makes the contributions of the boundary data and the non-boundary data to the prototypes reduced and enhanced,respectively.Furthermore,a weighting function is built to determine the weights of the data.The weighted data are used to modify the prototype matrix.With the modified prototype matrix,the partition matrix can be refined,which ultimately makes the boundaries of the information granules optimized.In the optimization design of degranulation mechanism,two new schemes also proposed.By building up a supervised learning mode of the granulation-degranulation based on the proposed matrix transformation model,and it is also further decomposed,then the partition matrix is optimized through a series of matrix operations.With the modified partition matrix,the original numeric data can be restored from the information granules which reduces the reconstruction error significantly.In addition,the degranulation mechanism is also enhanced by introducing a vector of fuzzification factors(fuzzification factor vector)and setting up an adjustment mechanism to modify the prototype matrix and the partition matrix.The design is regarded as an optimization problem,which is guided by a reconstruction criterion.In the proposed scheme,the initial partition matrix and prototype matrix are generated by the FCM.Then a fuzzification factor vector is defined to form an appropriate fuzzification factor for each cluster to build up an adjustment scheme of modifying the prototype matrix and the partition matrix.With the supervised learning mode of the granulation-degranulation process,a composite cost function of the fuzzification factor vector,the prototype matrix and the partition matrix is constructed.Subsequently,the Quantum-behaved Particle Swarm Optimization(QPSO)is employed to optimize the fuzzification factor vector to refine the prototype matrix and develop the optimal partition matrix.Finally,the reconstruction performance of the algorithm is enhanced.At the end of this research,the optimization of T-S model based on granulationdegranulation mechanism is also involved,that is,to optimize T-S model through building a series of reasonable information granules.Because from the point of view of granular computing,fuzzy models are considered as mappings from information granules expressed in the input and output spaces.A double fuzzy T-S model is designed based on the research of the granulation-degranulation mechanism.Compared with a scheme of other literature the excellent performance of the proposed scheme is verified.