This article introduces the Bayesian probabilistic fuzzy neural network (BPFNN), a unified architecture designed to overcome the challenges of conventional fuzzy clustering and neural networks in terms of uncertainty, noise, and interpretability. At its core, the Bayesian probabilistic fuzzy $C$ -means (BPFCMs) algorithm is employed to define the hidden-layer nodes, extending traditional FCM through non-Gaussian modeling and posterior inference via Markov chain Monte Carlo (MCMC). By combining Metropolis-Hastings (MHs) for membership updates with Gibbs sampling for parameter estimation, BPFCM yields probabilistic memberships that capture uncertainty in the antecedent rules more effectively than deterministic approaches. Since the hidden-layer activations represent only similarity values between inputs and cluster centers, the original input features are not directly preserved. To compensate, the hidden-to-output connections are formulated as linear functions of the input, ensuring recovery of discriminative information in the consequent rules. These functions are optimized using a generalized cross-entropy (GCE) objective, with iteratively reweighted least squares (IRLSs) employed for efficient and regularized updates. Extensive experiments on benchmark datasets and high-dimensional laser-induced breakdown spectroscopy (LIBS) spectral data confirm that BPFNN consistently surpasses both classical fuzzy systems and contemporary deep learning models, providing improved accuracy, robustness, and interpretability.

In this study, we propose a novel data transformation-driven fuzzy clustering neural network (DTFCNN) to enhance the dimensionality reduction function and end-to-end refinment learning ability of the entire structure. Unlike conventional fuzzy clustering-based neural networks, which rely on fuzzy c-means clustering in the hidden layer and least squares error-based learning for connection weights, the DTFCNN utilizes backpropagation (BP) learning as a refinement algorithm. This refinement process enables iterative fine-tuning of the model’s parameters, allowing it to adapt more effectively to complex patterns. By using BP, DTFCNN enhances its ability to learn intricate feature interactions and extract relevant features from high-dimensional spaces, thereby significantly improving the model’s flexibility and performance. The proposed DTFCNN consists of four layers. First, a preprocessing layer employs principal component analysis for feature extraction and dimensionality reduction, where eigenvectors are used as connection weights for the preprocessing layer. Second, a hidden layer utilizes fuzzy c-means clustering for generating fuzzy membership degrees, and centers serve as connection weights for hidden layers. The entries of the partition matrix also are regarded as membership degrees. In the output layer, a linear-driven affine transform is used for fitting connection weights, and a SoftMax function is employed to express the outputs as probabilities. Finally, all parameters, such as eigenvectors, centers, and coefficients, from the preprocessing layer to the output layer are refined through BP-based learning. To validate the effectiveness of the DTFCNN, we conducted a collection of comparison experiments: 1) publicly benchmark datasets with statistical analysis, 2) facial recognition datasets for application-specific testing, and 3) three large-scale datasets. The results demonstrate that the DTFCNN outperforms classical classifiers, state-of-the-art fuzzy classifiers, and deep learning baselines in terms of accuracy and generalization capability. The DTFCNN model uses fewer parameters than deep learning baselines, resulting in faster training times without compromising performance. Overall, the DTFCNN achieves higher accuracy while maintaining model adaptability.

Original Article: Robust Classification via Interval Type-2 Fuzzy C-Means and Gradient Boosting, IEEE Transactions on Fuzzy Systems, vol. 33, no. 9, pp. 3103–3117, Sept. 2025. doi:10.1109/TFUZZ.2025.3583051.

This article introduces the Bayesian probabilistic fuzzy neural network (BPFNN), designed to overcome the limitations of Fuzzy C-Means (FCM) clustering, which struggles with uncertainty, noise, nonlinearity, and interpretability. Additionally, it addresses the shortcomings of traditional objective functions, such as mean squared error (MSE), which fail to capture the complexities inherent in high-dimensional and uncertain datasets. The BPFNN framework integrates Bayesian probabilistic modeling with advanced fuzzy clustering techniques, utilizing a non-Gaussian probability density function to better represent data uncertainties. A hybrid Markov chain Monte Carlo strategy, combining Metropolis-Hastings for membership updates and Gibbs sampling for cluster parameter estimation, is employed to effectively model uncertainty. For the learning of connection weights, the generalized cross-entropy loss function is applied, and the iteratively reweighted least squares algorithm is used to update the weights, allowing for a more precise quantification of the divergence between predicted and ground truth labels. Experimental evaluations on several benchmark datasets, as well as a high-dimensional laser-induced breakdown spectroscopy (LIBS) spectral dataset, demonstrate that the proposed BPFNN significantly outperforms both traditional methods and State-of-the-Art techniques in terms of classification accuracy and robustness. Notably, BPFNN achieves an average accuracy improvement of 3.2% over conventional models on benchmark datasets, with a 5.3% improvement on the LIBS dataset, highlighting its substantial advancement in the field.

In this article, we present a dimensionality reduction methodology of reinforced fuzzy-rule-based neural networks (FRNNs) realized with the help of determination/correlation coefficient-based streamlined feature selection strategy and fuzzy clustering with standard deviation to cope with high-dimensional data. This approach aims to reduce the design process of the proposed networks and to curb the computational overhead inherently associated with the increasing volume of data both in terms of their number and the dimensionality of the feature space. The overall architecture and learning mechanism of the FRNNs are based on radial basis function neural networks. However, we design the hidden layer of RBFNNs differently by using fuzzy clustering, which makes it easy to determine the parameters, such as centers and widths of the receptive fields (activation functions). Unlike conventional neural networks, the RBFNNs do not have a feature to support dimensionality reduction. To overcome this limitation, FRNNs select input variables by evaluating the adjusted determination coefficient of the model. To reduce the computational burden of finding an appropriate combination of inputs, we propose a simplified feature selection and elimination technique, in which the variables are selected or eliminated by correlation coefficients. A linear function expresses the connection weight, and we apply L2-norm regularization to least-square-error-based learning to estimate stable coefficients (weights), which is expected to significantly improve the generalization ability. The superiority of the proposed FRNNs was demonstrated by using 28 real-world benchmark datasets. The networks are also compared with the conventional models associated with the FRNNs and several related models previously published in the literature.

This article is concerned with designing a tobacco leaves classifier through fuzzy clustering-based neural networks, which leverage multiple histogram analyses of images. The key issue of the study is to recognize high-quality and low-quality tobacco leaves only by using color images obtained from real industrial areas. This study applies multiple histogram analyses from different color spaces as image preprocessing to extract the meaningful features from high-resolution images. Dimensionality reduction is performed through principal component analysis to extract essential features to reduce model complexity and alleviate overfitting problems. In a classifier, we apply fuzzy clustering-based neural networks that incorporate fuzzy clustering techniques, especially fuzzy C-means clustering, along with a cross-entropy loss function and its learning mechanism. The process of setting and training the membership function of node in the hidden layer is substituted with fuzzy C-means clustering. Also, Softmax function produces the model's output in terms of class probabilities. The cost function of the networks is determined using the cross-entropy loss function, while the learning process involves Newton's method-based iterative nonlinear least square error estimation. The experiment validates the competitiveness of the proposed design methodology using real tobacco images obtained from the industry. The performance of the proposed classifier is compared against other classifiers previously reported in the literature to demonstrate its effectiveness.

In this article, we introduce a design methodology of reinforced fuzzy models based both on univariate analysis and multivariable analysis to cope with high-dimensional problems. This approach is aimed at reducing the design process and curbing computing overhead inherently associated with the increasing volume of data in terms of both their number and the dimensionality of the feature space. The critical features of the proposed fuzzy models are highlighted as follows: First, the essential input variables of the model are selected by running the univariable and multivariable analyses. In univariate analysis, input variables with a strong linear relationship with the output variable are selected through correlation analysis completed for each input space and output space. On the contrary, in multivariable analysis, input variables are chosen by comparing the determination coefficients obtained from the subsets of input variables. Second, according to the analysis of the input variable, we construct two different kinds of fuzzy models. The first fuzzy model comprises the design of the univariable-based fuzzy model (UFM) and its aggregation. The UFMs are made by the individual input variables selected from the univariable analysis using a correlation coefficient. The subspaces formed by correlation analysis are applied for determining the centers of the membership function (MF). The results produced by individual fuzzy models are aggregated through somet-conorms. The second fuzzy model is with the fuzzy clustering for improving the form of fuzzy space in the premise part of a fuzzy rule. To curb the dramatic increase in size of fuzzy rule in high-dimensional problems, the clustering space is employed as the fuzzy space, and the partition matrix produced by fuzzy provided the required degrees of the MF. Experimental studies include a suite of synthetic and publicly available data. The superiority of the proposed design methodology was demonstrated by using 34 publicly available datasets and also compared with the conventional models associated with the fuzzy rule-based models as well as the state-of-the-art models reported in the literature.

In this paper, we propose reinforced fuzzy clustering-based ensemble neural networks (FCENNs) classifier. The objective of this paper is focused on the development of the design methodologies of ensemble neural networks classifier for constructing the network structure and enhancing the learning methods of fuzzy clustering-based neural networks through the combination of the probabilistic model and its learning mechanism. The proposed FCENNs classifier takes into consideration a cross-entropy error function to improve learning while L2norm regularization is used to reduce overfitting as well as enhance generalization abilities. The essential points of the proposed reinforced FCENNs classifier can be enumerated as follows: First, in the proposed classifier, the cross-entropy error function is used as a cost function; to do this, a softmax function is applied to represent a categorical distribution located at the nodes of the output layer. Second, the learning mechanism is composed of two parts. First, fuzzy C-means clustering forms the connections (weights) of the hidden layer while the connections of the output layer are adjusted with the aid of the nonlinear least squares method using Newton's method-based learning. Third, L2norm-regularization is considered to avoid the degradation of generalization ability caused by overfitting. The learning mechanism similar to ridge regression is realized by adding L2penalty term to the cross-entropy error function. From the viewpoint of performance improvement achieved through the proposed novel learning method, the design methodology for the ensemble neural networks classifier is discussed and analyzed with the aid of a diversity of two-dimensional synthetic data and machine learning datasets.

This paper is concerned with a new design methodology of a reinforced interval type-2 fuzzy c-means (FCM) based fuzzy classifier (FC). The key point of this study is to reduce the computational complexity of type-2 fuzzy set-based models and to alleviate the deterioration of its generalization abilities through the synergistic effect of two algorithms: First, interval type-2 FCM (IT2FCM) is used in the hidden layer of the network and connections (weights) are adjusted by invoking the least squares error estimation method. Second, an L2-norm regularization is considered in the cost function to avoid the construction of the network suffering from overfitting. In more detail, the hidden layer of the proposed FC is realized by interval type-2 FCM clustering to deal with the factor of uncertainty involved in the problem. This type of clustering is realized by using two values of the fuzzification coefficient resulting in the interval type-2 membership functions. Once completing type reduction, the membership grades of IT2FCM are used as the outputs of the hidden layer. Instead of the backpropagation training, least squares estimator based learning is applied to adjust the functional connection being regarded as linear functions mapping the hidden layer to the output layer. In order to reduce potential overfitting, L2-norm regularization is taken into account. The effectiveness of the proposed classifier is analyzed with the aid of a number of machine learning datasets as well as face image datasets. Thorough comparative studies are also included.