This study concentrates on fuzzy relational calculus regarded as a basis of data compression. In this setting, images are represented as fuzzy relations. We investigate fuzzy relational equations as a basis of image compression. It is shown that both compression and decompression (reconstruction) phases are closely linked with the way in which fuzzy relational equations are being usually set and solved. The theoretical findings encountered in the theory of these equations are easily accommodated as a backbone of the relational compression. The character of the solutions to the equations make them ideal for reconstruction purposes as they specify the extremal elements of the solution set and in such a way help establish some envelopes of the original images under compression. The flexibility of the conceptual and algorithmic framework arising there is also discussed. Numerical examples provide a suitable illustrative material emphasizing the main features of the compression mechanisms.
In system modeling, knowledge management comes vividly into the picture when dealing with a collection of models. These models, being considered as sources of knowledge, are engaged in some collective pursuits of collaboration and consensus formation. We show the use of information granules in these processes by elaborating on their conceptual role. It is revealed that information granules are used to facilitate processes of collaboration and consensus building. Such granular constructs, referred to as granular models, can also emerge as a part of higher order models to reflect and quantify the diversity of the sources of knowledge involved in knowledge management. Several detailed algorithmic schemes are presented, along with related computational aspects associated with Granular Computing. It is also shown how the construction of information granules, through the use of the principle of justifiable granularity, becomes advantageous in the realization of granular models. This study builds upon seminal concepts established in L.A. Zadeh's Rosetta Stone paper devoted to information granulation.
The paper elaborates on the representation and reconstruction of numerical and nonnumerical data in fuzzy modeling. Proposed are general criteria leading to the distortion-free interfacing mechanisms that help transform information between the systems (or modeling environments) operating at different levels of information granularity. Distinguished are three basic categories of information: numerical, interval-valued, and linguistic (fuzzy). Since 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.
FCMs are aimed at modeling and simulation of dynamic systems. They exhibit numerous advantages, such as model transparency, simplicity, and adaptability to a given domain, to name a few. FCMs have been applied to numerous industrial and research areas. In some cases generic FCMs suffer from a certain drawback that originates from their definition and concerns a limited, first-order dynamics of processing realized at the nodes of the maps. In this study, we introduce a concept of higher-order memory based FCMs. The proposed extension modifies the simulation model of a generic FCM while it does not negatively impact transparency and simplicity of the model itself. We discuss several architectural alternatives along with the ensuing computing and optimization aspects. Preliminary experimental results included in this paper show superiority of the extended higher-order memory based FCMs over a generic FCM in terms of the modeling accuracy
With the remarkably diversified plethora of design methodologies and algorithmic pursuits present today in system modeling including fuzzy modeling, we also witness a surprisingly high level of homogeneity in the sense that the resulting models are predominantly concerned with and built by using a data set coming from a single data source. In this study, we introduce a concept of collaborative granular modeling. In a nutshell, we are faced with a number of separate sources of data and the resulting individual models formed on their basis. An ultimate objective is to realize modeling at the global basis by invoking effective mechanisms of knowledge sharing and collaboration. In this way, each model is formed not only by relying on a data set that becomes locally available but also is exposed to some general modeling perspective by effectively communicating with other models and sharing and reconciling revealed local sources of knowledge. Several fundamental modes of collaboration (by varying with respect to the levels of interaction) are investigated along with the concepts of collaboration mechanisms leading to the effective way of knowledge sharing and reconciling or calibrating the individual modeling points of view. The predominant role of information granules with this regard is stressed. For illustrative purposes, the underlying architecture of granular models investigated in this talk is concerned with rule-based topologies and rules of the form “if Ri then fi” with Ri being a certain information granule (typically set, fuzzy set or rough set) formed in the input space and fi denoting any local model realizing a certain mapping confined to the local region of the input space and specified by Ri. 1. INTRODUCTORY COMMENTS Fuzzy modeling (Angelov et al., 2008; Crtespo and Weber, 2005; Kacprzyk and Zadrozny, 2005; Kilic et al, 2007; Molina et al., 2006; Pedrycz and Gomide, 1998; Pham and Castellani, 2006) exhibits a surprisingly diversity of design methodologies. The concepts and architectures of neurofuzzy systems, evolutionary fuzzy systems are becoming more present in the literature. In spite of this variety, there is one very visible development aspect that cuts across the entire field of fuzzy modeling, that is fuzzy models are built on around a single data set. What becomes more apparent nowadays is a tendency of modeling a variety of distributed systems or phenomena, in which there are separate data sets, quite often quite remote in terms of location or distant in time. The same complex phenomenon could be perceived and modeled using different data sets collected individually and usually not shared. The data might be expressed in different feature spaces as the view at the process could be secured from different perspectives. The models developed individually could be treated as a multitude of sources of knowledge. Along with the individual design of fuzzy models, it could be beneficial to share sources of knowledge (models), reconcile findings, collaborate with intent of forming a model, which might offer a global, unified, comprehensive and holistic view at the underlying phenomenon. Under these Proceedings 24th European Conference on Modelling and Simulation ©ECMS Andrzej Bargiela, Sayed Azam Ali David Crowley, Eugene J.H. Kerckhoffs (Editors) ISBN: 978-0-9564944-0-5 / ISBN: 978-0-9564944-1-2 (CD) circumstances an effective way of knowledge sharing and reconciliation through a sound communication platform becomes of paramount relevance, see Figure 1. Figure 1: A General Platform of Knowledge Reconciliation and Collaboration in Fuzzy Modeling A situation portrayed in Figure 1 is shown in a somewhat general way not moving into the details. It is essential to note that the mechanisms of collaboration and reconciliation are realized through passing information granules rather than detailed numeric entities. The general category of fuzzy models under investigation embrace models described as a family of pairs , i=1, 2, ...,c. In essence, these pairs can be sought as concise representations of rules with Ri forming the condition part of the i-th rule and fi standing in the corresponding conclusion part. It is beneficial to emphasize that in such rules, we admit a genuine diversity of the local models formalized by fi. From the modeling perspective the expression fi(x,ai) could be literally any modeling construct, namely • fuzzy set, • linear or nonlinear regression function, • difference or differential equation, • finite state machine, • neural network One can cast the fuzzy models in a certain perspective by noting that by determining a collection of information granules (fuzzy sets) Ri, one establishes a certain view at the system/phenomenon. Subsequently, the conclusion parts (fi) are implied by the information granules and their detailed determination is realized once Ri have been fixed or further adjusted (refined). In light of the discussion on knowledge reconciliation and mechanisms of collaboration, it becomes apparent that the interaction can focus on information granules Ri and communication schemes that invoke exchange of granules whereas conclusion parts can be adjusted accordingly once the collaborative development of information granules has been completed. The main objectives of the study, that is reflected by the organization of the material, is to formulate and discuss a variety of collaborative models of fuzzy models as well as highlight the design principles. The constructs resulting through such collaboration give rise in one way or another to granular constructs of higher order, where the elevated level of granularity is a consequence of reconciliation of knowledge coming from the individual models. The principle of justifiable granularity is presented and shows how granularity emerges as a result of summarization of numeric information (and numeric membership values, in particular). A number of collaborative schemes are discussed where we identify main concepts and present some general ways in which such schemes can be realized. We also show how type-2 fuzzy sets (including interval-valued fuzzy sets) are formed as an immediate result of collaboration. Throughout this study, we adhere to the standard notation. In particular information granules – fuzzy sets are denoted by capital letters. The notation and terminology is the one being in common usage in the area of fuzzy sets. 2. THE PRINCIPLE OF JUSTIFIABLE
Fuzzy Cognitive Maps (FCMs) are a handy method for dynamic systems modeling. A recently introduced RCGA learning method allows for establishing high-quality FCMs from historical data. The main drawback of this method is its scalability. In this paper, a framework for a novel scalable learning method for FCMs is introduced. 1. Background
This study presents an idea of transistor level realization of the sherman learning algorithm of Self-Organizing Maps (SOMs) which is described in (4). The realization of this algorithm in hardware calls for a solution of several specic problems not present in software implementa- tion. The main problem is related to an iterative nature of the adaptation process of the neighboring neurons positioned at particular rings surround- ing the winning neuron. This makes the circuit structure of the SOM very complex. To come up with a feasible realization, we introduce some mod- ications to the original sherman algorithm. Detailed simulations of the software model of the SOM show that these modications do not have the negative impact on the learning process, and helps bring signicant reduc- tion of the circuit complexity. In consequence, a fully parallel adaptation of all neurons is possible, which makes the SOM very fast.
