Self-organizing neural networks with fuzzy polynomial neurons
Introduction
The challenging quest for constructing models of systems that come with significant approximation and generalization abilities as well as are easy to comprehend has been within the community for decades. While neural networks, fuzzy sets and evolutionary computing have augmented a field of modeling quite immensely, they have also given rise to a number of new methodological issues and increased awareness about tradeoffs one has to make in system modeling. When the dimensionality of the model goes up, so do the difficulties. In particular, when dealing with high-order non-linear and multivariable equations of the model, we require a vast amount of data for estimating all its parameters. The group method of data handling (GMDH) [1] is one of the approaches that help alleviate the problem. Fuzzy sets emphasized the aspect of transparency of the models and a role of a model designer whose prior knowledge about the system may be very helpful in facilitating all identification pursuits. On the other hand, to build models of high approximation capabilities, there is a need for advanced tools. The art of modeling is to reconcile these two tendencies and find a workable synergistic environment. In this paper, we study a new neurofuzzy topology, called a self-organizing neural network (SONN). SONN is a network resulting from the fusion of the extended GMDH algorithm and a fuzzy inference system. We introduce a complete learning scheme, discuss a way of growing the SONN and provide with a series of comprehensive experimental studies.
Section snippets
The SONN with FPNs
We introduce a fuzzy polynomial neuron (FPN). This neuron, regarded as a generic type of the processing unit, dwells on the concepts of fuzzy sets and neural networks. Fuzzy sets realize a linguistic interface by linking the external world—numeric data with the processing unit. Neurocomputing manifests in the form of a local polynomial unit realizing a non-linear processing. The FPN encapsulates a family of non-linear “if-then” rules. When arranged together, FPNs build a SONN. In the sequel, we
The topology of the SONN
Proceeding with the overall SONN architecture, see Fig. 2, essential design decisions have to be made with regard to the number of input variables and the order of the polynomial occurring in the conclusion part of the rule. Following these criteria, we distinguish between two fundamental types of the SONN architectures. Moreover, for each type of the topology we identify two cases:
- (a)
Basic SONN architecture—the number of the input variables of the fuzzy rules in the FPN node is kept the same in
The design of the SONN
The SONN comes with a highly versatile architecture both in the flexibility of the individual nodes (that are essentially banks of non-linear “if-then” rules) as well as the interconnectivity between the nodes and organization of the layers. Evidently, these features contribute to the significant flexibility of the networks yet require a prudent design methodology and a well-thought learning mechanisms. Let us stress that there are several important differences that make this architecture
Simulations
In this section, we demonstrate how SONN can be utilized to predict future values of a chaotic time series. The performance of the network is also contrasted with some other models existing in the literature. The time series is generated by the chaotic Mackey–Glass differential delay equation [2] of the form:The prediction of future values of this series arises as a benchmark problem that has been used and reported by a number of researchers. From the
Conclusions
In this paper, we have introduced an idea of a SONNs with FPNs, studied its properties and came up with a design procedure. Extensive experimental studies produced superb results. Some general observations can be summarized as follows: (i) with a properly selected type of membership functions and the organization of the layers, SONN performs better than other models; (ii) the architecture of the SONN is not fully predetermined and can be generated (adjusted) during learning; (iii) one can
Acknowledgements
Support from the Basic Research Program of the Korean Science and Engineering Foundation (KOSEF: Grant no. R02-2000-00284) and the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged.
References (7)
Polynomial theory of complex systems
IEEE Trans. Systems, Man Cybernetics
(1971)- et al.
Oscillation and chaos in physiological control systems
Science
(1977) - et al.
Generating fuzzy rules from numerical data, with applications
IEEE Trans. Systems, Man, Cybern.
(1992)
Cited by (39)
Design of stabilized polynomial-based ensemble fuzzy neural networks based on heterogeneous neurons and synergy of multiple techniques
2021, Information SciencesCitation Excerpt :There are lots of methods for synthesizing neural networks. Oh and Pedrycz combined fuzzy neural networks with polynomial neural networks [8] to propose a category of neuro fuzzy networks, called fuzzy polynomial neural networks (FPNNs) [9], and came up with a series of improvements and optimizations of FPNNs [10,11]. Park and Pedrycz designed an architecture of genetically oriented fuzzy relation neural networks and proposed a comprehensive design method to support the expansion of its network structure [12].
Self-organized hybrid fuzzy neural networks driven with the aid of probability-based node selection and enhanced input strategy
2020, NeurocomputingCitation Excerpt :Park et al. combined PNN with fuzzy neural network (FNN) and proposed a synergistic hybrid fuzzy modeling architecture, where PNN is taken as the consequence of the rules of the fuzzy model [10]. Oh and Pedrycz took Takagi-Sugeno-Kang (TSK) fuzzy models as nodes to replace polynomial nodes in PNN, and proposed a class of fuzzy polynomial neural networks [11,12]. In [13], the radial basis neural network is used as the new node of PNN, and the swarm intelligence algorithm is applied to optimize the parameters of the model.
Time series long-term forecasting model based on information granules and fuzzy clustering
2015, Engineering Applications of Artificial IntelligenceGranular computing neural-fuzzy modelling: A neutrosophic approach
2013, Applied Soft Computing JournalGenetically optimized Hybrid Fuzzy Set-based Polynomial Neural Networks
2011, Journal of the Franklin Institute