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Utilizing granular computing to enhance artificial neural network architecture, a newtype of network emerges--the granular neural network (GNN). GNNs offer distinct advantages over their traditional counterparts: The ability to process both numerical and granular data, leading to improved interpretability. This paper proposes a novel design method for constructing GNNs, drawing inspiration from existing interval-valued neural networks built upon NNNs. However, unlike the proposed algorithm in this work, which employs interval values or triangular fuzzy numbers for connections, existing methods rely on a pre-defined numerical network. This new method utilizes a uniform distribution of information granularity to granulate connections with unknown parameters, resulting in independent GNN structures. To quantify the granularity output of the network, the product of two common performance indices is adopted: The coverage of numerical data and the specificity of information granules. Optimizing this combined performance index helps determine the optimal parameters for the network. Finally, the paper presents the complete model construction and validates its feasibility through experiments on datasets from the UCIMachine Learning Repository. The results demonstrate the proposed algorithm's effectiveness and promising performance. [ABSTRACT FROM AUTHOR]

It is an interesting topic to interpret artificial neural networks (ANNs) by considering some change various approaches. This paper explores the relationship between the input and output units of the simplest ANN, a single layer perceptron for the binary classification problem, from the probability point of view. If the feature variables of datasets follow independent normal distribution and outputs are activated by sigmoid function or smooth Relu function, we advocate that the probability density function (pdf) of the output variable is an exponential family distribution. Furthermore, by introducing an intermediate variable, the pdf of the output variable can be written as a linear combination of three normal distributions with same spread but different centers. Based on these results, the probability of the predicted class label can be written as a standard normal cumulative distribution function (cdf). The originality of this paper comes with interesting theoretical results to provide ANNs with a new description of the relationship between input variables to output variables, which can enable ANNs to be understood from a new perspective. Extensive experiments based on one artificial synthesized dataset and ten real‐world benchmark datasets validate the reasonability of those results. [ABSTRACT FROM AUTHOR]

As one of the most essential sources of energy, byproduct gas plays a pivotal role in the steel industry, for which the flow tendency is generally regarded as the guidance for planning and scheduling in real production. In order to obtain the numeric estimation along with its reliability, the construction of prediction intervals (PIs) is highly demanded by any practical applications as well as being long term for providing more information on future trends. Bearing this in mind, in this article, a hierarchical granular computing (HGrC)-based model is established for constructing long-term PIs, in which probabilistic modeling gives rise to a long horizon of numeric prediction, and the deployment of information granularities hierarchically extends the result to be interval-valued format. Considering that the structure of this model has a direct impact on its performance, Monte-Carlo search with a policy gradient technique is then applied for reinforcement structure learning. Compared with the existing methods, the size (length) of the granules in the proposed approach is unequal so that it becomes effective for not only periodic but also nonperiodic data. Furthermore, with the use of parallel strategy, the efficiency can be also guaranteed for real-world applications. The experimental results demonstrate that the proposed method is superior to other commonly encountered techniques, and the stability of the structure learning process behaves better when compared with other reinforcement learning approaches. [ABSTRACT FROM AUTHOR]

Granular data streams (GDSs) are a class of high-level abstract multitime scale description of data streams. Prediction intervals (PIs) for GDSs that provide estimated values as well as their corresponding reliability play an important role for assisting on-site workers to perceive the nonstationary environment in real time. However, constructing reliable PIs for GDSs constitutes a significant challenge. To provide a solution to the problem, an interval type-2 (IT2) fuzzy granular neural network (FGNN) dynamic ensemble approach (IT2FGNNDEnsemble) is proposed in this article. To fully reflect the uncertainty of GDSs, an interval value learning algorithm based IT2FGNN is developed, which can automatically generate, prune, merge, and realize recall in a single-pass learning mode. In addition, an evolving dynamic ensemble method is presented by providing an adaptive structure that considers a tradeoff between coverage and width of PIs, which can dynamically generate and prune the element of an ensemble according to current data tendency. A number of synthetic and industrial data streams experimentally validate the performance of the proposed IT2FGNNDEnsemble by using the state-of-the-art comparative methods. It is demonstrated that the proposed approach exhibits a good performance on PIs for practical applications. [ABSTRACT FROM AUTHOR]

Neural networks (NNs) with least square error (LSE) estimation form a certain type of single hidden layer feed-forward NNs. In this class of networks, the input connections (weights) and the biases of hidden neurons are generated randomly and fixed after being generated. The output connections are estimated by the LSE method rather than the back-propagation method. The random generation of the input connection weights and the hidden biases results in the larger number of hidden neurons to assure the quality of classification performance. To reduce the number of neurons in the hidden layer while maintaining the classification performance, we apply a “divide and conquer” strategy in this article. In other words, we divide an overall input space into several subspaces by using information granulation technique (Fuzzy C-Means clustering algorithm) and determine the local decision boundaries among related subspaces. A decision boundary defined in the input space can be considered as being composed of several decision boundaries defined in subspaces that form the entire input space. For the decision boundaries defined in the subspaces, their nonlinearity becomes lower in comparison with the one being encountered when considering the entire input space. Through the weighted LSE estimation instead of using the LSE estimation method, the connections of several NNs can be estimated without interfering with each other. After estimating the weights, the decision boundaries defined in the related subspaces are merged to a single decision boundary by using fuzzy ensemble technique. Several machine learning datasets and one real world application dataset are used to evaluate and validate the proposed fuzzy ensemble classifier. Based on the experimental results, the proposed classifier shows better classification performance when compared with the performance of some selected classifiers. [ABSTRACT FROM AUTHOR]

We present a new approach to multiple time series (MTS) prediction using many-inputs many-outputs (MIMO) fuzzy aggregation models (FAM) with modular neural networks (MNNs). We propose different FAM to generate the forecast outcome of MTS. Several representative approaches to MIMO-FAM are considered. The first FAM is designed with the use of adaptive neuro-fuzzy inference systems (ANFIS) using subtractive clustering (referred to as ANFIS-SC) and fuzzy C-means (referred to as ANFIS-FCM). The second FAM is developed based on Type-1 fuzzy inference systems, while the third FAM exploits interval Type-2 fuzzy inference systems. We design different MNN architectures, and the learning is carried out using backpropagation algorithms. The MTS used in the experiments concerned publicly available data including the Mexican Stock Exchange, National Association of Securities Dealers Automated Quotation and Taiwan Stock Exchange time series. The FAM are compared on the basis of the prediction errors based on commonly used performance indexes, such as the mean absolute error, the mean square error and the root-mean-square error. Simulation results demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we develop a granular input space for neural networks, especially for multilayer perceptrons (MLPs). Unlike conventional neural networks, a neural network with granular input is an augmented study on a basis of a well learned numeric neural network. We explore an efficient way of forming granular input variables so that the corresponding granular outputs of the neural network achieve the highest values of the criteria of specificity (and support). When we augment neural networks through distributing information granularities across input variables, the output of a network has different levels of sensitivity on different input variables. Capturing the relationship between input variables and output result becomes of a great help for mining knowledge from the data. And in this way, important features of the data can be easily found. As an essential design asset, information granules are considered in this construct. The quantification of information granules is viewed as levels of granularity which is given by the expert. The detailed optimization procedure of allocation of information granularity is realized by an improved partheno genetic algorithm (IPGA). The proposed algorithm is testified effective by some numeric studies completed for synthetic data and data coming from the machine learning and StatLib repositories. Moreover, the experimental studies offer a deep insight into the specificity of input features. Highlights • An algorithm of developing a granular neural network with granular input on basis of a designed network is proposed. • The influence of different levels of information granularity on the performance of the granular network is studied. • An improved partheno genetic algorithm is used to optimize the allocation of information granularity. • Synthetic data and real world data sets are used to testify the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

As large amounts of plastics are widely used in diverse areas of industry, the amount of plastic waste, including black plastics, continues to increase. In this situation, the necessity of useful recycling having limited resources gradually increases. The design of plastic classification systems for plastics recycling becomes more important to effectively address recycling activities. Until now, conventional sorting systems based on the near infrared ray technology have been used to classify plastic wastes. However, the classification of black plastic waste still remains a challenge because such materials do not reflect sufficient signals due to the absorption of laser light coming from the NIR spectrometer. In order to solve such problems, this research is focused on an efficient way to identify black plastics. Attenuated Total Reflectance (ATR) Fourier Transform Infrared Radiation (FT-IR) and a Raman spectrometer are used to carry out qualitative and quantitative analysis for the effective as well as efficient classification of black plastic wastes. In this study, to effectively classify the black plastic waste, data processing and Fuzzy Transform (F-Transform) as well as PCA-based Fuzzy Radial Basis Function Neural Networks (FRBFNNs) classifier is proposed. Input variables extracted on a basis of chemical characteristic peaks as well as interval range positioned near the chemical characteristic peaks were exploited as a way to improve the classification performance of the FRBFNN classifier. In order to evaluate the performance of the classifier, a suite of techniques including F-Transform-based as well as Principal Component Analysis (PCA)-based FRBFNNs classifier designed with the aid of Particle Swam Optimization are developed to analyze and classify black plastics. [ABSTRACT FROM AUTHOR]

In this study, we present a comprehensible classifier AFSNN that embeds a new type of coherence membership function, which builds upon the theoretical findings of the axiomatic fuzzy set (AFS) theory into the hidden layer of neural network with random weights (NNRWs). Borrowing from the idea of NNRWs that employs the random initialization technique, the relation among attributes, simple concepts, and complex concepts are randomly determined. Complex concepts are generated through the combination of randomly selected simple concepts by AFS logic operation. The output weights of NNRWs are utilized to evaluate the confidence of each complex concept for every target class, which means that the feasibility of complex concepts for every class is determined analytically rather than through the tuning parameters of constraint conditions such as in conventional AFS-based classifiers. For the proposed method, compared to other neural-network-based classification methods, the fuzzy descriptions generated from complex concepts in hidden layer make classification result human understandable. We have experimented with several benchmark datasets and compared the results with other neural network-based classifiers. We show that our method outperforms Ensemble, EvRBFN, NNEP, LVQ, and iRProp+ in the seven out of ten datasets. The results show that the performance of AFSNN is competitive in terms of classification accuracy and the network shows a distinctive capability of providing explicit knowledge in the form of linguistic description. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

In this study, we propose a concept of fuzzy wavelet polynomial neural networks (FWPNNs) based on concepts and constructs of polynomial neural networks and fuzzy wavelet neurons (FWNs). These networks exhibit a rule-based architecture while each rule in the FWN consists of the premise part and consequence part. The premise part is realized by using C-means clustering method, while the consequence part is realized by means of wavelet functions whose parameters are estimated with the aid of the least square method. In some sense, the FWPNN can be regarded as a generalized fuzzy wavelet neural network (FWNN). Unlike Gaussian membership functions that are commonly utilized to implement the premise part of the rules in typical FWNNs, C-means method is employed here to overcome a possible curse of dimensionality. Polynomial neural networks (PNNs) are used to express the nonlinearity of a complex system. Furthermore, the particle swarm optimization is used to optimize the design parameters of the proposed network. Based on the PNNs and FWNNs, the proposed FWPNNs take advantages of these two neural networks: it exhibits the abilities to describe high-order nonlinear relations between input and output variables and it is beneficial to describe models impacted by uncertainty. The proposed FWPNNs are applied for time-series prediction and regression problems (e.g., control of dynamic plants). Several well-known modeling benchmarks including regression and time series are considered to evaluate the performance of the proposed FWPNNs. A comparative analysis shows that the proposed FWPNNs result in better performance when comparing with some previous models reported in the literature. [ABSTRACT FROM PUBLISHER]

In this paper, we introduce a new architecture of optimized Radial Basis Function neural network classifier developed with the aid of fuzzy clustering and data preprocessing techniques and discuss its comprehensive design methodology. In the preprocessing part, the Linear Discriminant Analysis (LDA) or Principal Component Analysis (PCA) algorithm forms a front end of the network. The transformed data produced here are used as the inputs of the network. In the premise part, the Fuzzy C-Means (FCM) algorithm determines the receptive field associated with the condition part of the rules. The connection weights of the classifier are of functional nature and come as polynomial functions forming the consequent part. The Particle Swarm Optimization algorithm optimizes a number of essential parameters needed to improve the accuracy of the classifier. Those optimized parameters include the type of data preprocessing, the dimensionality of the feature vectors produced by the LDA (or PCA), the number of clusters (rules), the fuzzification coefficient used in the FCM algorithm and the orders of the polynomials of networks. The performance of the proposed classifier is reported for several benchmarking data-sets and is compared with the performance of other classifiers reported in the previous studies. [ABSTRACT FROM PUBLISHER]

Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems. [ABSTRACT FROM AUTHOR]

Clustering has been applied to numerous areas, including signal and image processing. Many approaches have been developed over the years to efficiently construct granular models on a basis of numerical experimental data. In this study, we propose a novel approach to construct a granular model that is fundamentally designed around information granules regarded as hyperboxes. Several studies have been focused on building a set of hyperboxes around data; one of them being a Min-Max Neural Network (NN) algorithm. Here we develop two different methods to construct these information granules, nevertheless some essential similarities to previous studies can be found. In particular, hyperboxes are constructed by using some reference data, and they are endowed with some parametric flexibility to facilitate controlling their size, whereas the construction of the hyperboxes involve elimination or reduction of possible overlaps between them. In the proposed approach, given a set of input and output data pairs, we construct interval-based information granules to partition the output space (viz. the space of the output variable). On a basis of these intervals, we carry out a so-called context-based Fuzzy C-Means algorithm to construct cluster centers (prototypes) in the multivariable input space. These prototypes serve as hyperbox cores. To construct the information granules, two methods are studied: one develops a family of hyperboxes by realizing some constrictions, while the other one engages Differential Evolution (DE) to realize further optimization. To reduce overlap, two methods are tested: one being previously proposed for the min-max NN and a new one, which engages DE to optimize the overlap reduction. Experimental studies involve synthetic data and publicly available real-world data. The results are compared with the outcomes produced by the algorithm proposed by Simpson. The performance of the method is quantified and it is demonstrated that the obtained results are substantially better when dealing with multi-dimensional data. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new topology and offer a comprehensive design methodology of fuzzy set-based neural networks (FsNNs). The proposed architecture of the FsNNs is based on the fuzzy polynomial neurons formed through a collection of 'if-then' fuzzy rules, fuzzy inference, and polynomials with the extended structure of the premise and the consequence parts of fuzzy rules being formed within the networks. Three different forms of regression polynomials (namely constant, linear, and quadratic) are used in the consequence part of the rules. In order to build an optimal FsNN, the underlying structural and parametric optimization is supported by a dynamic search-based genetic algorithm (GA), which forms an optimal solution through successive adjustments (refinements) of the search range. The structure optimization involves the determination of the input variables included in the premise part and the order of the polynomial forming the consequence part of the rules. In the study, we explore two types of optimization methodologies, namely a simultaneous tuning and a separate tuning. GAs are global optimizers; however, when being used in their generic version, they often lead to a significant computing overhead caused by the need to explore an excessively large search space. To eliminate this shortcoming and increase the effectiveness of the optimization itself, we introduce a dynamic search-based GA that results in a rapid convergence while narrowing down the search to a limited region of the search space. We exploit this optimization mechanism to be completed both at the structural as well as the parametric level. To evaluate the performance of the proposed FsNN, we offer a suite of several representative numerical examples. [ABSTRACT FROM AUTHOR]

The paper presents a concept of an air pollution monitoring system with prediction abilities, based on wireless smart sensors, that takes into account local conditions (microclimate) prevailing in particular areas of the city. In most cases reported in the literature, artificial neural networks (ANNs) are used to predict future pollution levels. In existing solutions of this type, ANNs are trained with generalized datasets common for larger areas, e.g., cities. Our investigations show, however, that conditions may strongly differ even between particular streets in the city, which may impact prediction quality. This results from varying density of urban development, different levels of insolation, airiness, amounts of greenery, etc. As a result, with similar values of ANN input signals, such as current pollution levels, temperature, pressure, etc., the results of the prediction may differ significantly from reality. For this reason, we propose an innovative solution, in which particular sensors are equipped with miniaturized low-power ANNs, trained with datasets gathered directly from their closest environment, without a need for the obtaining of such data from a base station. This may simplify the installation and maintenance process of a network of such sensors. In a further part of this work, we dealt with solutions that enable the reduction of the computational complexity of ANNs in the case of their implementation on specialized integrated circuits. We propose replacing the most complex mathematical operations used in the learning algorithm with simpler solutions. A prototype chip containing the main blocks of such an ANN was also designed. [ABSTRACT FROM AUTHOR]

This study provides a general introduction to the principles, algorithms and practice of computational intelligence (CI) and elaborates on those facets with relation to biomedical signal analysis, especially ECG signals. We discuss the main technologies of computational intelligence (namely, neural networks, fuzzy sets or granular computing, and evolutionary optimization), identify their focal points and stress an overall synergistic character, which ultimately gives rise to the highly symbiotic CI environment. Furthermore, the main advantages and limitations of the CI technologies are discussed. In the sequel, we present CI-oriented constructs in signal modeling, classification, and interpretation. Examples of the CI-based ECG signal processing problems are presented. [ABSTRACT FROM AUTHOR]

In this study, we introduce a new topology of radial basis function-based polynomial neural networks (RPNNs) that is based on a genetically optimized multi-layer perceptron with radial polynomial neurons (RPNs). This paper offers a comprehensive design methodology involving various mechanisms of optimization, especially fuzzy C-means (FCM) clustering and particle swarm optimization (PSO). In contrast to the typical architectures encountered in polynomial neural networks (PNNs), our main objective is to develop a topology and establish a comprehensive design strategy of RPNNs: (a) The architecture of the proposed network consists of radial polynomial neurons (RPN). These neurons are fully reflective of the structure encountered in numeric data, which are granulated with the aid of FCM clustering. RPN dwells on the concepts of a collection of radial basis function and the function-based nonlinear polynomial processing. (b) The PSO-based design procedure being applied to each layer of the RPNN leads to the selection of preferred nodes of the network whose local parameters (such as the number of input variables, a collection of the specific subset of input variables, the order of the polynomial, the number of clusters of FCM clustering, and a fuzzification coefficient of the FCM method) are properly adjusted. The performance of the RPNN is quantified through a series of experiments where we use several modeling benchmarks, namely a synthetic three-dimensional data and learning machine data (computer hardware data, abalone data, MPG data, and Boston housing data) already used in neuro-fuzzy modeling. A comparative analysis shows that the proposed RPNN exhibits higher accuracy in comparison with some previous models available in the literature. [ABSTRACT FROM AUTHOR]

We present a new programmable neighborhood mechanism for hardware implemented Kohonen self-organizing maps (SOMs) with three different map topologies realized on a single chip. The proposed circuit comes as a fully parallel and asynchronous architecture. The mechanism is very fast. In a medium sized map with several hundreds neurons implemented in the complementary metal-oxide semiconductor 0.18 \mum technology, all neurons start adapting the weights after no more than 11 ns. The adaptation is then carried out in parallel. This is an evident advantage in comparison with the commonly used software-realized SOMs. The circuit is robust against the process, supply voltage and environment temperature variations. Due to a simple structure, it features low energy consumption of a few pJ per neuron per a single learning pattern. In this paper, we discuss different aspects of hardware realization, such as a suitable selection of the map topology and the initial neighborhood range, as the optimization of these parameters is essential when looking from the circuit complexity point of view. For the optimal values of these parameters, the chip area and the power dissipation can be reduced even by 60% and 80%, respectively, without affecting the quality of learning. [ABSTRACT FROM PUBLISHER]

Abstract: In this study, we are concerned with a construction of granular neural networks (GNNs)—architectures formed as a direct result reconciliation of results produced by a collection of local neural networks constructed on a basis of individual data sets. Being cognizant of the diversity of the results produced by the collection of networks, we arrive at the concept of granular neural network, producing results in the form of information granules (rather than plain numeric entities) that become reflective of the diversity of the results generated by the contributing networks. The design of a granular neural network exploits the concept of justifiable granularity. Introduced is a performance index quantifying the quality of information granules generated by the granular neural network. This study is illustrated with the aid of machine learning data sets. The experimental results provide a detailed insight into the developed granular neural networks. [Copyright &y& Elsevier]

Neural networks (NNs) implemented at the transistor level are powerful adaptive systems. They can perform hundreds of operations in parallel but at the expense of a large number of building blocks. In the case of analog realization, an extremely low chip area and low power dissipation can be achieved. To accomplish this, the building blocks should be simple. This brief presents a new current-mode low-complexity flexible adaptive mechanism (ADM) with a strongly reduced leakage in analog memory. Input signals ranging from 0.5 to 20 \mu\A are held for 10–50 ms, with the leakage rate from 0.2%/ms to 0.04%/ms, respectively, depending on temperature. A small storage capacitor of 200 fF enables a short write time ( < 100 ns). A single ADM cell occupies 1400 \mu\m^2 when realized in the Taiwan Semiconductor Manufacturing Company Ltd. CMOS 0.18-\mu\m technology. The potential application of this NN is envisioned in a mobile platform based on a wireless sensor network to be used for online analysis of electrocardiography signals. [ABSTRACT FROM PUBLISHER]

This paper presents a complementary metal-oxide-semiconductor (CMOS) implementation of a conscience mechanism used to improve the effectiveness of learning in the winner-takes-all (WTA) artificial neural networks (ANNs) realized at the transistor level. This mechanism makes it possible to eliminate the effect of the so-called "dead neurons," which do not take part in the learning phase competition. These neurons usually have a detrimental effect on the network performance, increasing the quantization error. The proposed mechanism comes as part of the analog implementation of the WTA neural networks (NNs) designed for applications to ultralow power portable diagnostic devices for online analysis of ECG biomedical signals. The study presents Matlab simulations of the network's model, discusses postlayout circuit level simulations and includes results of measurement completed for the physical realization of the circuit. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a new category of current-mode łukasiewicz OR and AND logic neurons and ensuing logic networks along with their ultra-low power realization. The introduced circuits can operate in a wide range of the input signals varying in-between 10nA and 10μA. For low current values the operating point of transistors is set in the under threshold region. In this region, the mismatch between transistors exhibits a far stronger impact on the current mirror precision than the one observed in case of the strong inversion region. The proposed design alleviates this problem by reducing the number of current mirrors between the input and the output of the neuron and of the overall network to only one. łukasiewicz operators require only summation and subtraction operations, which make them suitable for realization in analog current-mode technique. In this case even large number of input signals can be summed in a simple junction in a single step. This is the reason of choosing łukasiewicz operations in the proposed circuit. Using other t-norm and t-conorm operations with multiplication and division operations would make the realization of the circuit very difficult and inefficient. [Copyright &y& Elsevier]

Abstract: The ultimate challenges of system modeling concern designing accurate yet highly transparent and user-centric models. We have witnessed a plethora of neurofuzzy architectures which are aimed at addressing these two highly conflicting requirements. This study is concerned with the design and the development of transparent logic networks realized with the aid of fuzzy neurons and fuzzy unineurons. The construction of networks of this form requires a formation of efficient interfaces that constitute a conceptually appealing bridge between the model and the real-world experimental environment in which the model is to be used. In general, the interfaces are constructed by invoking some form of granulation of information; and binary (Boolean) discretization, in particular. We introduce a new discretization environment that is realized by means of particle swarm optimization (PSO) and data clustering implemented by the K-Means algorithm. The underlying structure of the network is optimized by invoking a combination of the PSO and the mechanisms of conventional gradient-based learning. We discuss various optimization strategies by considering Boolean as well as fuzzy data coming as the result of discretization of original experimental data and then involving several learning strategies. We elaborate on the interpretation aspects of the network and show how those could be strengthened through efficient pruning. We also show how the interpreted network leads to a simpler and more accurate logic description of the experimental data. A number of experimental studies are included. [Copyright &y& Elsevier]

Abstract: In this study, we concentrate on the fundamentals and essential development issues of logic-driven constructs of fuzzy neural networks. These networks, referred to as logic-oriented neural networks, constitute an interesting conceptual and computational framework that greatly benefits from the establishment of highly synergistic links between the technology of fuzzy sets (or granular computing, being more general) and neural networks. The most essential advantages of the proposed networks are twofold. First, the transparency of neural architectures becomes highly relevant when dealing with the mechanisms of efficient learning. Here the learning is augmented by the fact that domain knowledge could be easily incorporated in advance prior to any learning. This becomes possible given the compatibility between the architecture of the problem and the induced topology of the neural network. Second, once the training has been completed, the network can be easily interpreted and thus it directly translates into a series of truth-quantifiable logic expressions formed over a collection of information granules. The design process of the logic networks synergistically exploits the principles of information granulation, logic computing and underlying optimization including those biologically inspired techniques (such as particle swarm optimization, genetic algorithms and alike). We elaborate on the existing development trends, present key methodological pursuits and algorithms. In particular, we show how the logic blueprint of the networks is supported by the use of various constructs of fuzzy sets including logic operators, logic neurons, referential operators and fuzzy relational constructs. [Copyright &y& Elsevier]

In this study, we present a new architecture of a granular neural network and provide a comprehensive design methodology as well as elaborate on an algorithmic setup supporting its development. The proposed neural network relates to a broad category of radial basis function neural networks (RBFNNs) in the sense that its topology involves a collection of receptive fields. In contrast to the standard architectures encountered in RBFNNs, here we form individual receptive fields in subspaces of the original input space rather than in the entire input space. These subspaces could be different for different receptive fields. The architecture of the network is fully reflective of the structure encountered in the training data which are granulated with the aid of clustering techniques. More specifically, the output space is granulated with use of K-means clustering while the information granules in the multidimensional input space are formed by using the so-called context-based fuzzy C-means, which takes into account the structure being already formed in the output space. The innovative development facet of the network involves a dynamic reduction of dimensionality of the input space in which the information granules are formed in the subspace of the overall input space which is formed by selecting a suitable subset of input variables so that this subspace retains the structure of the entire space. As this search is of combinatorial character, we use the technique of genetic optimization [genetic algorithms (GAs), to be more specific] to determine the optimal input subspaces. A series of numeric studies exploiting synthetic data and data coming from the Machine Learning Repository, University of California at Irvine, provide a detailed insight into the nature of the algorithm and its parameters as well as offer some comparative analysis. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a new neural network architecture based on a family of referential multilayer perceptrons (RMLPs) that play a role of generalized receptive fields. In contrast to “standard” radial basis function (RBF) neural networks, the proposed topology of the network offers a considerable level of flexibility as the resulting receptive fields are highly diversified and capable of adjusting themselves to the characteristics of the locally available experimental data. We discuss in detail a design strategy of the novel architecture that fully exploits the modeling capabilities of the contributing RMLPs. The strategy comprises three phases. In the first phase, we form a “blueprint” of the network by employing a specialized version of the commonly encountered fuzzy C-means (FCM) clustering algorithm, namely the conditional (context-based) FCM. In this phase our intent is to generate a collection of information granules (fuzzy sets) in the space of input and output variables, narrowed down to some certain contexts. In the second phase, based upon a global view at the structure, we refine the input–output relationships by engaging a collection of RMLPs where each RMLP is trained by using the subset of data associated with the corresponding context fuzzy set. During training each receptive field focuses on the characteristics of these locally available data and builds a nonlinear mapping in a referential mode. Finally, the connections of the receptive fields are optimized through global minimization of the linear aggregation unit located at the output layer of the overall architecture. We also include a series of numeric experiments involving synthetic and real-world data sets which provide a thorough comparative analysis with standard RBF neural networks. [Copyright &y& Elsevier]

Abstract: A fuzzy inference system (FIS) architecture based on the Takagi–Sugeno–Kang (TSK) fuzzy model is developed for time series prediction. Our objective is to investigate and evaluate the proposed rule-based model against commonly used time series models including “standard” architectures such as autoregressive (AR) models and selected topologies of neural networks. The main architectural developments of the FIS involve fuzzy relational antecedents (viz., antecedents represented in the form of fuzzy relations) and recurrent neural networks forming the consequents of the rules. Fuzzy C-means (FCM) clustering is applied to the time series to determine the fuzzy relations for the antecedents of the rules. Experimental results are reported for single-time step prediction and multiple time step (p-step) prediction on several widely used time series. [Copyright &y& Elsevier]

In this study, we introduce an advanced architecture of genetically optimized Hybrid Fuzzy Neural Networks (gHFNN) and develop a comprehensive design methodology supporting their construction. A series of numeric experiments is included to illustrate the performance of the networks. The construction of gHFNN exploits fundamental technologies of Computational Intelligence (CI), namely fuzzy sets, neural networks, and genetic algorithms (GAs). The architecture of the gHFNNs results from a synergistic usage of the genetic optimization-driven hybrid system generated by combining Fuzzy Neural Networks (FNN) with Polynomial Neural Networks (PNN). In this tandem, a FNN supports the formation of the condition part of the rule-based structure of the gHFNN. The conclusion part of the gHFNN is designed using PNNs. We distinguish between two types of the simplified fuzzy inference rule-based FNN structures showing how this taxonomy depends upon the type of a fuzzy partition of input variables. As to the conclusion part of the gHFNN, the development of the PNN dwells on two general optimization mechanisms: the structural optimization is realized via GAs whereas in case of the parametric optimization we proceed with a standard least square method-based learning. To evaluate the performance of the gHFNN, we experimented with three representative numerical examples. A comparative analysis demonstrates that the proposed gHFNN come with higher accuracy as well as superb predictive capabilities when compared with other neurofuzzy models. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a new architecture of information granulation-based and genetically optimized Hybrid Self-Organizing Fuzzy Polynomial Neural Networks (HSOFPNN). Such networks are based on genetically optimized multi-layer perceptrons. We develop their comprehensive design methodology involving mechanisms of genetic optimization and information granulation. The architecture of the resulting HSOFPNN combines fuzzy polynomial neurons (FPNs) that are located at the first layer of the network with polynomial neurons (PNs) forming the remaining layers of the network. The augmented version of the HSOFPNN, “IG_gHSOFPNN”, for brief, embraces the concept of information granulation and subsequently exhibits higher level of flexibility and leads to simpler architectures and rapid convergence speed to optimal structure in comparison with the HSOFPNNs and SOFPNNs. The GA-based design procedure being applied at each layer of HSOFPNN leads to the selection of preferred nodes of the network (FPNs or PNs) whose local characteristics (such as the number of input variables, the order of the polynomial, a collection of the specific subset of input variables, the number of membership functions for each input variable, and the type of membership function) can be easily adjusted. In the sequel, two general optimization mechanisms are explored. The structural optimization is realized via GAs whereas the ensuing detailed parametric optimization is afterwards carried out in the setting of a standard least square method-based learning. The obtained results demonstrate a superiority of the proposed networks over the existing fuzzy and neural models. [Copyright &y& Elsevier]

Fuzzy neurons may have outstanding learning abilities and are endowed with significant interpretation capabilities. In this study, we are concerned with the development of logic networks composed of fuzzy neurons. The main phase of the design includes the granulation of the output space (via triangular fuzzy sets) being realized with the use of fuzzy equalization. In the sequel these fuzzy sets are used to guide the construction of a family of fuzzy sets in the input space. Further processing of the resulting fuzzy sets deals with some additional aggregation of those that are not sufficiently distinct. This helps reduce the size of the logic network. We include comprehensive experimentation and offer a thorough interpretation of the networks. Experiments concerning real-world continuous data help evaluate the network's appealing properties: transparent interpretability and practical feasibility. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1249–1267, 2006. [ABSTRACT FROM AUTHOR]

In this paper, the authors propose and investigate a new category of neurofuzzy networks-fuzzy polynomial neural networks (FPNNs)—and develop a comprehensive design methodology involving mechanisms of genetic optimization and, in particular, genetic algorithms (GAs). The conventional FPNNs developed so far are based on mechanisms of self-organization and evolutionary optimization. The design of the network exploits the extended group method of data handling, with some essential parameters of the network being provided by the designer and kept fixed throughout the overall development process. This restriction may hamper a possibility of producing an optimal architecture of the model. The design proposed in this paper addresses this issue. The augmented genetically optimized FPNN (gFPNN) results in a structurally optimized structure and comes with a higher level of flexibility in comparison with the one encountered in the conventional FPNN. The GA-based design procedure that is applied to each layer of FPNN leads to the selection of the preferred nodes (or fuzzy polynomial neurons) available within the FPNN. In the sequel, two general optimization mechanisms are explored. First, the structural optimization is realized via GAs, whereas the ensuing, detailed parametric optimization is carried out in the setting of a standard least-square-method-based learning. The performance of gFPNN is quantified through experimentation where a number of modeling benchmarks are being used, i.e., synthetic and experimental data already experimented within fuzzy or neurofuzzy modeling. The obtained results demonstrate superiority of the proposed networks over the existing fuzzy and neural models. [ABSTRACT FROM AUTHOR]

Abstract: In this study, we introduce a new architecture of hybrid fuzzy neural networks (gHFNNs) and offer a comprehensive design methodology that supports their development. The gHFNN rule-based architecture results from a synergistic usage of Fuzzy Neural Networks (FNNs) with Polynomial Neural Networks (PNNs). The FNN contributes to the formation of the premise part of the overall network of the gHFNN. The consequence part of the gHFNN is designed taking advantage of PNNs. The optimization of the FNN is realized with the aid of a standard back-propagation learning combined with genetic optimization. The development of the PNN dwells on the extended Group Method of Data Handling (GMDH) and Genetic Algorithms (GAs). Through the consecutive process of such structural and parametric optimization, an optimized topology of the PNN becomes generated in a dynamic fashion. The performance of the gHFNN is evaluated through a series of numeric experiments. A comparative analysis shows that the proposed gHFNN is characterized by higher accuracy as well as significant predictive capabilities when contrasted with other neurofuzzy models presented in the literature. [Copyright &y& Elsevier]

In this paper, the fundamental idea of linguistic models introduced by Pedrycz and Vasilakos (1999) is followed and their comprehensive design framework is developed. The paradigm of linguistic modeling is concerned with constructing models that: 1) are user centric and 2) inherently dwell upon collections of highly interpretable and user-oriented entities such as information granules. The objective of this paper is to investigate and compare alternative design options, present an organization of the overall optimization process, and come up with a specification of several evaluation mechanisms of the performance of the models. The underlying design tool guiding the development of linguistic models revolves around the augmented version of fuzzy clustering known as a context-based or conditional fuzzy C-means (C-FCM). The design process comprises several main phases such as: 1) defining and further refining context fuzzy sets; 2) completing conditional fuzzy clustering; and 3) optimizing parameters (connections) linking information granules in the input and output spaces. An iterative process of forming information granules in the input and output spaces is discussed. Their membership functions are adjusted by the gradient-based learning guided by the minimization of some performance index. The paper comes with a comprehensive suite of experiments that lead to some design guidelines of the models. Furthermore, the performance of linguistic models is contrasted with that of other fuzzy models, especially radial basis function neural networks (RBFNNs) and related constructs that are based on concepts of fuzzy clustering. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new topology of fuzzy polynomial neural networks (FPNNs) that is based on a genetically optimized multilayer perceptron with fuzzy polynomial neurons (FPNs). The study offers a comprehensive design methodology involving mechanisms of genetic optimization, especially those exploiting genetic algorithms (GAs). Let us recall that the design of the ‘conventional’ FPNNs uses an extended group method of data handling (GMDH) and uses a fixed scheme of fuzzy inference (such as simplified, linear, and regression polynomial fuzzy inference) in each FPN of the network. It also considers a fixed number of input nodes (as being selected in advance by a network designer) at FPNs (or nodes) located in each layer. However such design process does not guarantee that the resulting FPNs will always result in an optimal networks architecture. Here, the development of the FPNN gives rise to a structurally optimized topology and comes with a substantial level of flexibility which becomes apparent when contrasted with the one we encounter in the conventional FPNNs. The design of each layer of the FPNN deals with its structural optimization involving a selection of preferred nodes (or FPNs) with specific local characteristics (such as the number of input variables, the order of the polynomial forming a consequent part of fuzzy rules and a collection of the specific subset of input variables) and addresses detailed aspects of parametric optimization. Along this line, two general optimization mechanisms are explored. The structural optimization is realized via GAs. In case of the parametric optimization we proceed with a standard least square method-based learning. Through the consecutive process of such structural and parametric optimization, an optimized and flexible fuzzy neural network becomes generated in a dynamic fashion. To evaluate the performance of the genetically optimized FPNN (gFPNN), we experimented with two time series data (gas furnace and chaotic time series) as well as some synthetic data. A comparative analysis reveals that the proposed FPNN exhibits higher accuracy and superb predictive capability in comparison to some previous models available in the literature. [ABSTRACT FROM AUTHOR]

Clustering with partial supervision finds its application in Situations where data is neither entirely nor accurately labeled. This paper discusses a semi-supervised clustering algorithm based on a modified version of the fuzzy C-Means (FCM) algorithm. The objective function of the proposed algorithm consists of two components. The first concerns traditional unsupervised clustering while the second tracks the relationship between classes (available labels) and the clusters generated by the first component. The balance between the two components is tuned by a scaling factor. Comprehensive experimental studies are presented. First, the discrimination of the proposed algorithm is discussed before its reformulation as a classifier is addressed. The induced classifier is evaluated on completely labeled data and validated by comparison against some fully supervised classifiers, namely support vector machines and neural networks. This classifier is then evaluated and compared against three semi-supervised algorithms in the context of learning from partly labeled data. In addition, the behavior of the algorithm is discussed and the relation between classes and clusters is investigated using a linear regression model. Finally, the complexity of the algorithm is briefly discussed. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a new architecture of hybrid fuzzy polynomial neural networks (HFPNN) that is based on a genetically optimized multi-layer perceptron and develop their comprehensive design methodology involving mechanisms of genetic optimization. The construction of HFPNN exploits fundamental technologies of computational intelligence (CI), namely fuzzy sets, neural networks, and genetic algorithms (GAs). The architecture of the resulting genetically optimized HFPNN (namely gHFPNN) results from a synergistic usage of the hybrid system generated by combining fuzzy polynomial neurons (FPNs)-based fuzzy neural networks (FNN) with polynomial neurons (PNs)-based polynomial neural networks (PNN). The design of the conventional HFPNN exploits the extended group method of data handling (GMDH) with some essential parameters of the network being provided by the designer and kept fixed throughout the overall development process. This restriction may hamper a possibility of producing an optimal architecture of the model. The augmented gHFPNN results in a structurally optimized structure and comes with a higher level of flexibility in comparison to the one we encounter in the conventional HFPNN. The GA-based design procedure being applied at each layer of HFPNN leads to the selection of preferred nodes (FPNs or PNs) available within the HFPNN. In the sequel, two general optimization mechanisms are explored. First, the structural optimization is realized via GAs whereas the ensuing detailed parametric optimization is carried out in the setting of a standard least square method-based learning. The performance of the gHFPNN is quantified through experimentation where we exploit data coming from processes of pH neutralization and NOx emission. These datasets have already been used quite intensively in fuzzy and neurofuzzy modeling. The obtained results demonstrate superiority of the proposed networks over the existing fuzzy and neural models. [Copyright &y& Elsevier]

The recent trend in the development of neurofuzzy systems has profoundly emphasized the importance of synergy between the fundamentals of fuzzy sets and neural networks. The resulting frameworks of the neurofuzzy systems took advantage of an array of learning mechanisms primarily originating within the theory of neurocomputing and the use of fuzzy models (predominantly rule-based systems) being well established in the realm of fuzzy sets. Ideally, one can anticipate that neurofuzzy systems should fully exploit the linkages between these two technologies while strongly preserving their evident identities (plasticity or learning abilities to be shared by the transparency and full interpretability of the resulting neurofuzzy constructs). Interestingly, this synergy still becomes a target yet to be satisfied. This study is an attempt to address the fundamental interpretability challenge of neurofuzzy systems. Our underlying conjecture is that the transparency of any neurofuzzy system links directly with the logic fabric of the system so the logic fundamentals of the underlying architecture become of primordial relevance. Having this in mind the development of neurofuzzy models hinges on a collection of logic driven processing units named here fuzzy (logic) neurons. These are conceptually simple logic-oriented elements that come with a well-defined semantics and plasticity. Owing to their diversity, such neurons form essential building blocks of the networks. The study revisits the existing categories of logic neurons, provides with their taxonomy, helps understand their functional features and sheds light on their behavior when being treated as computational components of any neurofuzzy architecture. The two main categories of aggregative and reference neurons are deeply rooted in the fundamental operations encountered in the technology of fuzzy sets (including logic operations, linguistic modifiers, and logic reference operations). The developed heterogeneous networks come with a well-defined semantics and high interpretability (which directly translates into the rule-based representation of the networks). As the network takes advantage of various logic neurons, this imposes an immediate requirement of structural optimization, which in this study is addressed by utilizing various mechanisms of genetic optimization (genetic algorithms). We discuss the development of the networks, elaborate on the interpretation aspects and include a number of illustrative numeric examples. [ABSTRACT FROM AUTHOR]

In this paper, we develop a method for recognizing face images by combining wavelet decomposition, Fisherface method, and fuzzy integral. The proposed approach is comprised of four main stages. The first stage uses the wavelet decomposition that helps extract Intrinsic features of face images. As a result of this decomposition, we obtain four subimages (namely approximation, horizontal, vertical, and diagonal detailed images). The second stage of the approach concerns the application of the Fisherface method to these four decompositions. The choice of the Fisherface method in this setting is motivated by its insensitivity to large variation in light direction, face pose, and facial expression. The two last phases are concerned with the aggregation of the individual classifiers by means of the fuzzy integral. Both Sugeno and Choquet type of fuzzy integral are considered as the aggregalion method. In the experiments we use n-fold cross-validation to assure high consistency of the produced classification outcomes. The experimental results obtained for the Chungbuk National University (CNU) and Yale University face databases reveal that the approach presented in this paper yields better classification performance in comparison to the results obtained by other classifiers. [ABSTRACT FROM AUTHOR]

Experimental software data sets describing software projects in terms of their complexity and development time have been a subject of intensive modeling. A number of various modeling methodologies and modeling designs have been proposed including neural networks, fuzzy, and neurofuzzy models. In this study, we introduce a concept of Self-organizing neurofuzzy networks (SONFN), a hybrid modeling architecture combining neurofuzzy networks (NFN) and polynomial neural networks (PNN). The development of the SONFN dwells on the technologies of Computational Intelligence (CI), namely fuzzy sets, neural networks, and genetic algorithms. The architecture of the SONFN results from a synergistic usage of neurofuzzy networks (NFNs) and polynomial neural networks (PNNs). NFNs contribute to the formation of the premise part of the rule-based structure of the SONFN. The consequence part of the SONFN is designed using PNNs. We discuss two classes of SONFN architectures and propose comprehensive learning algorithms. The experimental results include well-known software data such as the NASA data set concerning software cost estimation and the one describing software modules of the Medical Imaging System (MIS). [Copyright &y& Elsevier]

In this study we consider the classification (supervised learning) problem in [0 1]^n that utilizes fuzzy sets as pattern classes. Each class is described by one or more fuzzy hyperbox defined by their corresponding minimum- and maximum vertices and the hyperbox membership function. Two types of hyperboxes are created: inclusion hyperboxes that contain input patterns belonging to the same class, and exclusion hyperboxes that contain patterns belonging to two or more classes, thus representing contentious areas of the pattern space. With these two types of hyperboxes each class fuzzy set is represented as a union of inclusion hyperboxes of the same class minus a union of exclusion hyperboxes. The subtraction of sets provides for efficient representation of complex topologies of pattern classes without resorting to a large number of small hyperboxes to describe each class. The proposed fuzzy hyperbox classification is compared to the original Min-Max Neural Network and the Gene ral Fuzzy Min-Max Neural Network and the origins of the improved performance of the proposed classification are identified. These are verified on a standard data set from the Machine Learning Repository. [ABSTRACT FROM AUTHOR]

The recent trend in the development of neurofuzzy systems has profoundly emphasized the importance of synergy between the fundamentals of fuzzy sets and neural networks. The resulting frameworks of the neurofuzzy systems took advantage of an array of learning mechanisms primarily originating within the theory of neurocomputing and the use of fuzzy models (predominantly rule-based systems) being well established in the realm of fuzzy sets. Ideally, one can anticipate that neurofuzzy systems should fully exploit the linkages between these two technologies while strongly preserving their evident identities (plasticity or learning abilities to be shared by the transparency and full interpretability of the resulting neurofuzzy constructs). Interestingly, this synergy still becomes a target yet to be satisfied. This study is an attempt to address the fundamental interpretability challenge we encountered in neurofuzzy systems. Our underlying conjecture is that the transparency of any neurofuzzy system links directly with the logic fabric of the system, so the logic fundamentals of the underlying architecture become of primordial relevance. Having this in mind, the development of neurofuzzy models hinges on a collection of logic driven processing units named here fuzzy (logic) neurons. These are conceptually simple logicoriented elements that come with a well-defined semantics and plasticity. Owing to their diversity, such neurons form essential building blocks of the networks. The study revisits the existing categories of logic neurons, provides with their taxonomy, helps understand their functional features and sheds light on their behavior when being treated as computational components of any neurofuzzy architecture. The two main categories of aggregative and reference neurons are deeply rooted in the fundamental operations encountered in the technology of fuzzy sets (including logic operations, linguistic modifiers, and logic reference operations). The developed heterogeneous networks come with a well-defined semantics and high interpretability (which directly translates into the rule-based representation of the networks). As the network takes advantage of various logic neurons, this imposes an immediate requirement of structural optimization, which in this study is addressed by utilizing various mechanisms of genetic optimization (genetic algorithms). We discuss the development of the networks, elaborate on the interpretation aspects and include a number of illustrative numeric examples. [ABSTRACT FROM AUTHOR]

In this study, we introduce and investigate a class of neural architectures of polynomial neural networks (PNNs), develop a comprehensive design methodology and carry out a series of numeric experiments. PNN is a flexible neural architecture whose structure (topology) is developed through learning. In particular, the number of layers of the PNN is not fixed in advance but is generated on the fly. In this sense, PNN is a self-organizing network. We distinguish between two kinds of SOPNN architectures, that is, (a) polynomial neuron (PN) based and (b) fuzzy polynomial neuron (FPN) based self-organizing networks. This taxonomy is based on the character of each neuron structure in the network. Each of them comes with two structures referred here to as basic and the modified topology. Moreover, for each topology of the SOPNN we identify two types that is a generic and advanced type. The essence of the design procedure of self-organizing polynomial neural networks (SOPNN) dwells on the group method of data handling (GMDH) (IEEE Trans. Systems Man and Cybernet. 12 (1971) 364). Each node of the PN based SOPNN exhibits a high level of flexibility and realizes a polynomial type of mapping (linear, quadratic, and cubic) between input and output variables. FPN based SOPNN dwells on the ideas of fuzzy rule-based computing and neural networks. Especially in FPN based SOPNN, the generic rules in the system assume the form “if A then y = P(x)” where A is a fuzzy relation in the condition space while P(x) is a polynomial forming a conclusion part of the rule. Each FPN (processing element) consists of a series of the nonlinear inference rules. The conclusion part of the rules, especially the regression polynomial uses several types of high-order polynomials such as linear, quadratic, modified quadratic, and cubic. As the premise part of the rules, both triangular and Gaussian-like membership functions are studied. Simulations involve a series of synthetic as well as experimental data used across various neurofuzzy systems. A detailed comparative analysis is included as well. [Copyright &y& Elsevier]

In this paper, we introduce a category of Multi-Fuzzy-Neural Networks (Multi-FNNs) models, analyze the underlying architectures and propose a comprehensive identification framework. The proposed Multi-FNNs are based on a concept of fuzzy rule-based FNNs that use H ard C-M eans (HCM) clustering and evolutionary fuzzy granulation and exploit linear inference being treated as a generic inference mechanism of approximate reasoning. By this nature, this FNN model is geared toward capturing relationships between information granules–fuzzy sets. The form of the information granules themselves (in particular their distribution and a type of membership functions) becomes an important design feature of the FNN model that contributes to its structural and parametric optimization. The genetically guided global optimization is then augmented by more refined gradient-based learning mechanisms such as a standard back-propagation. The HCM algorithm, whose role is to carry out preprocessing of the experimental data, is utilized to determine the structure of Multi-FNNs. The detailed parameters of the Multi-FNN (such as apexes of membership functions, learning rates, and momentum coefficients) are adjusted using genetic algorithms. The proposed aggregate performance index helps achieve a sound balance between approximation and generalization (predictive) abilities of the model. To evaluate an effectiveness of the introduced model, several numeric data sets are experimented with. Those include a time-series data of gas furnace, NOx emission process of gas turbine power plant and some synthetic data. [Copyright &y& Elsevier]