Radial Basis Function Centers Research Articles

Generalized [formula omitted]-sparse algorithms for constructing fault tolerant RBF networks

In the construction process of radial basis function (RBF) networks, two common crucial issues arise: the selection of RBF centers and the effective utilization of the given source without encountering the overfitting problem. Another important issue is the fault tolerant capability. That is, when noise or faults exist in a trained network, it is crucial that the network’s performance does not undergo significant deterioration or decrease. However, without employing a fault tolerant procedure, a trained RBF network may exhibit significantly poor performance. Unfortunately, most existing algorithms are unable to simultaneously address all of the aforementioned issues. This paper proposes fault tolerant training algorithms that can simultaneously select RBF nodes and train RBF output weights. Additionally, our algorithms can directly control the number of RBF nodes in an explicit manner, eliminating the need for a time-consuming procedure to tune the regularization parameter and achieve the target RBF network size. Based on simulation results, our algorithms demonstrate improved test set performance when more RBF nodes are used, effectively utilizing the given source without encountering the overfitting problem. This paper first defines a fault tolerant objective function, which includes a term to suppress the effects of weight faults and weight noise. This term also prevents the issue of overfitting, resulting in better test set performance when more RBF nodes are utilized. With the defined objective function, the training process is designed to solve a generalized M-sparse problem by incorporating an ℓ0-norm constraint. The ℓ0-norm constraint allows us to directly and explicitly control the number of RBF nodes. To address the generalized M-sparse problem, we introduce the noise-resistant iterative hard thresholding (NR-IHT) algorithm. The convergence properties of the NR-IHT algorithm are subsequently discussed theoretically. To further enhance performance, we incorporate the momentum concept into the NR-IHT algorithm, referring to the modified version as “NR-IHT-Mom”. Simulation results show that both the NR-IHT algorithm and the NR-IHT-Mom algorithm outperform several state-of-the-art comparison algorithms.

Extreme learning machine–radial basis function neural network–based state-of-charge estimation of lithium-ion batteries assisted with fiber Bragg grating sensor measurements

Lithium-ion batteries have been widely used in power systems for accepting large-scale renewable energy generations and in the transport sector to facilitate vehicle decarbonization, the accuracy of state-of-charge (SOC) estimation is extremely crucial for the battery systems to operate safely and reliably. Many approaches have been developed for battery SOC estimation based on the measurable electrical signals. However, the electrical measurements are usually corrupted by sensor noise and are also limited by insulation, corrosion, and electromagnetic compatibility. To address these problems, fiber Bragg grating (FBG) sensors are used to replace thermocouple sensors, the battery SOC therefore can be estimated based on the measurement of the strain information. In order to train the SOC estimation model for lithium-ion batteries, an FBG sensor–based battery SOC estimator, namely, ELM-RBFNN, is developed using the radial basis function neural network (RBFNN) identified by extreme learning machine (ELM). According to the ELM principle, the ELM-RBFNN model randomly selects the radial basis function (RBF) centers and determines the parameters of the hidden-layer nodes without any manual adjustment. The ELM-RBFNN estimation model has the advantages of fast learning speed, global optimization, and excellent generalization. Experimental results confirm that compared to the linear estimator and the polynomial estimator, the developed SOC estimator is able to produce more accurate results.

A Brand-New Simple, Fast, and Effective Residual-Based Method for Radial Basis Function Neural Networks Training

The radial basis function (RBF) neural network is a type of universal approximator, and has been widely used in various fields. Improving the training speed and compactness of RBF networks are critical for promoting their applications. In the present study, we propose a simple, fast, and effective RBF networks training method, which is based on the residual extreme points and their neighborhoods (thus called the REN method for short in this paper). The REN method calculates RBF centers and widths through a two-level iterative process, and realizes two main functionalities, namely 1) adding multiple centers within one pass through the whole data set, and 2) calculating RBF widths specifically for each center. The use of this algorithm does not need any parameter adjustments, and the models for approximation or classification can be obtained by only one run. The performance of the proposed REN algorithm is compared with the classic and powerful orthogonal least squares (OLS) algorithm. By reaching the same accuracies, the REN algorithm trains RBF networks 50 and 320 times faster, in the chirp (0˜50 Hz, 2 s, 1 kHz, 2001 samples) and two-dimensional peaks (2401 samples) signal approximation tasks respectively, than the OLS algorithm does, and the number of centers obtained by the REN algorithm is reduced by half. When incorporating the same number of centers, the REN algorithm achieves accuracies up to 3 orders of magnitude higher than the best results obtained by the OLS algorithm. In the classification task of a real discrete breast cancer data, both methods result in accuracies comparable to many existent methods, but the REN algorithm has the advantages of fast training speeds and no requirements for parameter adjustments. The REN algorithm proposed in this study may potentially be used for tasks with large scale of data or applications that require high model performances.

Design of Iterative Fuzzy Radial Basis Function Neural Networks Based on Iterative Weighted Fuzzy C-Means Clustering and Weighted LSE Estimation

In this article, a reinforced iterative fuzzy radial basis function neural networks (IFRBFNN) is introduced as an augmented FRBFNN architecture generated through the iterative refinement process of weighted fuzzy C-means (WFCM) clustering and weighted least square error estimation. It is well known that the location of the clusters have an effect on the classification performance of the FRBFNN. The underlying idea behind this article is how to define the center points of clusters based on the data distribution analysis as well as the improvement of classification performance. In a nutshell, while the fuzzy C-means clustering to define the positions of radial basis functions (RBFs) based in the unsupervised learning manner is usually used in a FRBFNN, the parameters (cluster centers and their ensuing polynomial coefficients) refinement of FRBFNN through the proposed iterative method lead to superb classification performance by relocating the positions of RBFs over the input space implied by supervised learning. The idea of the proposed approach is to relocate the centers (prototypes) of the fuzzy clusters by using WFCM clustering algorithm and to re-estimate the coefficients by weighted least square estimation with the aid of the cross-entropy loss values of data so that the classification performance is improved. The weight related to each data is determined by some auxiliary information (i.e., the modified version of the cross-entropy loss value of each data). When it comes to the estimation of the coefficients of the consequent polynomial in IFRBFNN, they are estimated by the weighted least square error estimation technique, where the weights of the coefficients are defined based on the cross-entropy loss values. The auxiliary information such as the weights for relocation of the centers of RBFs and the weights for the re-estimation of the polynomial coefficients should be defined from the viewpoint of enhancement of the classification performance. The cross-entropy loss value of each data is able to meet the necessity of the required supervision signal. Several numerical experiments are provided to demonstrate the usefulness of the proposed design method for the FRBFNN for classification problems.

An enhanced artificial bee colony algorithm using radial bases function neural networks for color image segmentation

The aim of segmentation process is to partition the image into homogeneous regions or objects. The proposed segmentation algorithms using adaptive feature weighting to distinguish the significance of various features. To extract effective feature extraction could be used for establishing regions in region based image segmentation. The main goal of this exploration work is to segment objects that separating the foreground and background of the color images. The segmentation for color images is focused on the Enhanced Artificial Bee Colony segmentation algorithm that utilizes the Radial Basis Function (EABCRBF) neural network algorithm. This work arranges RBF neural networks to predict the end limits of the segmentation, which are formed from the artificial bee colony province made in the image histogram topography. The radial basis function initial parameters, for instance, centers and widths are automatically set upon the histogram peaks and minima. The proposed EABCRBF strategy is compared with two existing techniques of Artificial Bee Colony (ABC) and Fuzzy C-Means (FCM). The EABCRBF method gives great outcomes for color image segmentation as it extracts additional information from color images. The experimentation results are shown through Mat Lab R2013a.

New Strategy Based on RBF Network to Develop a Collaborative Filtering Recommender System

Collaborative filtering is a popular recommendation algorithm. It predicts user's interests according to the ratings or behaviour of other users in the system. However, the collaborative filtering recommender system suffers from several major limitations including scalability, sparsity, and cold start. In this paper, a collaborative filtering recommendation approach using radial basis function (RBF) network and power method is proposed. The proposed system has offline and online phases. In the offline phase, the sparse user-item rating matrix is completed by using RBF network based on Cover's theorem on the separability of patterns. RBF network learning is done by unsupervised kernel-based fuzzy c-means clustering algorithm for selecting RBF centers, and supervised gradient descend method for selecting RBF weights. In the offline phase, we predict non-rated items of a user. Then the full rating matrix is used to rank all the users. The ranking is done by solving an eigenvalue problem. This paper overcomes the scalability problem by clustering the users, the sparsity problem by completing the sparse rating matrix, and the new user cold start problem by recommending the top rated items of the high-ranked user. The results of the experiments, on the benchmark data sets, show that the proposed system produces high quality recommendation, in terms of accuracy and quality.

Construction and Simulation of Failure Distribution Model for Cycloidal Gears Grinding Machine

To increase reliability, reduce machine tool failure, and shorten maintenance time, serval failure distribution models are discussed based on the cycloidal gears form grinding. A fault data expansion algorithm model based on the radial basis function (RBF) neural network is proposed to accurately evaluate the reliability of cycloidal gears form grinding machines. The model uses a self-organizing clustering learning algorithm to determine the RBF centers and expansion constants of the neural network. It trains the neural network using the learning algorithm’s output and imports the randomly generated cumulative failure distribution function to obtain simulation data. A numerical example of researching failure distribution is presented which shows that the estimated value of mean time between failures (MTBF) is 909.20h, and the estimated value of reliability is 0.4874. Finally, the fault tree analysis-analytic hierarchy process (FTA-AHP) is used to analyze the fault tree of the cycloidal gear grinding machine. The reliability evaluation indexes are obtained by analyzing the corresponding reliability characteristic function, which proves the failure distribution model is valid. It has important guiding significance in evaluating the reliability and performance of large computer numerical control (CNC) form gear grinding machines.

Radial basis functions method for nonlinear time- and space-fractional Fokker-Planck equation

In this study, a radial basis functions (RBFs) method for solving nonlinear timeand space-fractional Fokker-Planck equation is presented. The time-fractional derivative is of the Caputo type, and the space-fractional derivatives are considered in the sense of Caputo or Riemann-Liouville. The Caputo and Riemann-Liouville fractional derivatives of RBFs are computed and utilized for approximating the spatial fractional derivatives of the unknown function. Also, in each time step, the time-fractional derivative is approximated by the high order formulas introduced in [6], and then a collocation method is applied. The centers of RBFs are chosen as suitable collocation points. Thus, in each time step, the computations of fractional Fokker-Planck equation are reduced to a nonlinear system of algebraic equations. Several numerical examples are included to demonstrate the applicability, accuracy, and stability of the method. Numerical experiments show that the experimental order of convergence is 4 − α where α is the order of time derivative.

Predicting the Compressive Strength of Concrete Using an RBF-ANN Model

In this study, a radial basis function (RBF) artificial neural network (ANN) model for predicting the 28-day compressive strength of concrete is established. The database used in this study is the expansion by adding data from other works to the one used in the author’s previous work. The stochastic gradient approach presented in the textbook is employed for determining the centers of RBFs and their shape parameters. With an extremely large number of training iterations and just a few RBFs in the ANN, all the RBF-ANNs have converged to the solutions of global minimum error. So, the only consideration of whether the ANN can work in practical uses is just the issue of over-fitting. The ANN with only three RBFs is finally chosen. The results of verification imply that the present RBF-ANN model outperforms the BP-ANN model in the author’s previous work. The centers of the RBFs, their shape parameters, their weights, and the threshold are all listed in this article. With these numbers and using the formulae expressed in this article, anyone can predict the 28-day compressive strength of concrete according to the concrete mix proportioning on his/her own.

A Unified Meshfree Pseudospectral Method for Solving Both Classical and Fractional PDEs

In this paper, we propose a meshfree method based on the Gaussian radial basis function (RBF) to solve both classical and fractional PDEs. The proposed method takes advantage of the analytical Laplacian of Gaussian functions so as to accommodate the discretization of the classical and fractional Laplacian in a single framework and avoid the large computational cost for numerical evaluation of the fractional derivatives. These important merits distinguish it from other numerical methods for fractional PDEs. Moreover, our method is simple and easy to handle complex geometry and local refinement, and its computer program implementation remains the same for any dimension $d \ge 1$. Extensive numerical experiments are provided to study the performance of our method in both approximating the Dirichlet Laplace operators and solving PDE problems. Compared to the recently proposed Wendland RBF method, our method exactly incorporates the Dirichlet boundary conditions into the scheme and is free of the Gibbs phenomenon as observed in the literature. Our studies suggest that to obtain good accuracy the shape parameter cannot be too small or too big, and the optimal shape parameter might depend on the RBF center points and the solution properties.

Constrained PSO Based Center Selection for RBF Networks Under Concurrent Fault Situation

In the training of radial basis function (RBF) networks, one important issue is to select RBF centers before constructing the networks. Most existing center selection methods are designed for the fault-free situation only. However, as the implementation of the networks may be perturbed by faults, these algorithms may lead to networks with degraded performance. This paper considers the center selection problem for RBF networks under the concurrent fault situation where multiplicative weight noise and open weight fault exist simultaneously. In particular, we introduce a binary label vector indicating the centers selected from training samples. Using the label vector, the fault-tolerant RBF model under the concurrent fault situation is reformulated as a constrained optimization problem, so that fault-tolerance can be considered in the procedure of center selection. To solve this constrained optimization problem, a constrained particle swarm optimization based algorithm is developed to select centers and train the network simultaneously. Simulation results show that the proposed algorithm is superior than state-of-the-art center selection algorithms.

Orthogonal least squares based center selection for fault-tolerant RBF networks

In the construction of a radial basis function (RBF) network, one of the most important issues is to determine the number of RBF nodes in the hidden layer and the corresponding centers. However, most existing center selection methods for RBF networks are designed for the faultless implementation situation only. In practice, the implementation of the network may be perturbed by faults and thus fault tolerance should be considered in selecting centers. This paper addresses this problem by focusing on the center selection problem of fault-tolerant RBF networks. In particular, we consider RBF networks under the concurrent fault situation and reformulate the fault-tolerant training objective function as a linear regression problem. A center selection and learning algorithm is then proposed based on the orthogonal least squares (OLS) algorithm. The proposed algorithm takes fault-tolerance into consideration both in center selection and network training, and it is able to determine the number of RBF nodes, the RBF centers and the trained connecting weights simultaneously. Simulation results show that the proposed algorithm is superior than state-of-the-art center selection algorithms.

A comparison of fixed- and free-positioned point mass methods for regional gravity field modeling

Radial basis functions (RBFs) have been used widely for regional gravity field modeling. By using the RBFs with the point mass kernel, the RBF-based technique becomes the point mass method. The model setup of the point mass RBFs, consisting of the selection of the spectral bandwidths, depths and network, is crucial for the quality of recovered regional gravity field models. If the spectral bandwidths are defined, the point mass method can be classified into the fixed- and free-positioned methods. This study compares the two methods for modeling of regional gravity fields in a unified framework. The fixed-positioned method uses a Reuter grid to determine the RBF number and centers. The grid depth is chosen by experimenting several candidates to achieve the smallest root-mean-square (RMS) of the differences between predicted and observed values at a set of control points. The magnitudes of the RBFs are estimated by solving a linear equation system, where Tikhonov regularization is applied if the normal matrix is ill-conditioned. The free-positioned method starts with initially unknown positions of the RBFs, and also the number of RBFs is determined in the computation process. Here, we use a search process to select the RBFs automatically by means of solving a series of nonlinear problems with depth constraints on the RBFs to minimize the RMS difference between the predictions and observations. The magnitudes of all selected RBFs are later re-estimated in the least-squares sense while keeping their positions unchanged. EGM2008 coefficients are firstly used to simulate two harmonic fields to test the performance of the two methods on various Reuter grids, depth limits, and spectral bandwidths, in order to gain certain guidelines for a proper selection of the model parameters. The two methods are then applied to real gravity data sets in the Auvergne and White Sands test areas, respectively. The results reveal that the free-positioned method outperforms the fixed-positioned method in regions with rough gravity field features while using less RBFs. In regions with smooth gravity field features, both methods give similar results, where the fixed-positioned method needs more RBFs than the free-positioned method.

Удосконалена інформаційна технологія для моніторингу зворотного гортанного нерва

The information technology has been proposed and substantiated to detect the location of the laryngeal nerve in the surgical wound during operations on the thyroid gland. The information technology, in contrast to existing, reduces the risk of the laryngeal nerve damage and the time of operation on the thyroid gland due to the developed interval model with radial basis functions. The mathematical models, which basis functions are radial, have a simple generalized structure and their structural identification is reduced to calculating the number of centers of radial basis functions. The main characteristics of the surgical wound tissue during surgery on the thyroid gland are its type and the distance from the point of irritation of the surgical area to the laryngeal nerve. As a result of research and processing of information signals of different patients, indicators of the type of the surgical wound tissue are substantiated. Such characteristics are the spectral components with the maximum amplitude. The patient information signals resulting from the response to muscle and nerve irritation at different distances to the laryngeal nerve have been analyzed. The information signals of patients, which respond to the irritation of the surgical wound with alternating current at equal distances to the laryngeal nerve of different patients, are similar, but the characteristics of the signals are different. The interval data analysis methods to eliminate the uncertainty and heterogeneity of the data sample have been proposed A clustering algorithm of inhomogeneous data samples has been used, which is based on the integration of similar spectral characteristics of information signals into groups described at intervals. The true distance from the irritation point of the surgical area to the back laryngeal nerve is within the constructed distance interval. This approach makes it possible to construct a single mathematical model based on radial basis functions to predict the distance from the irritation point to the laryngeal nerve. The results of the experiments confirmed the efficiency of advanced information technology for the laryngeal nerve monitoring

An $\ell_0$ -Norm-Based Centers Selection for Failure Tolerant RBF Networks

There are two important issues in the construction of a radial basis function (RBF) neural network. The first one is to select suitable RBF centers. The second one is that the resultant RBF network should be with good fault tolerance. This paper proposes an algorithm that is able to select RBF centers and to train fault tolerant RBF networks simultaneously. The proposed algorithm borrows the concept from sparse approximation. In our formulation, we first define a fault tolerant objective function based on all input vectors from the training samples. We then introduce the minimax concave penalty (MCP) function, which is an approximation of l 0 -norm, into the objective function. The MCP term is able to force some unimportant RBF weights to zero. Hence the RBF node selection process can be achieved during training. As the MCP function is nondifferentiable and nonconvex, traditional gradient descent based algorithms are still unable to minimize the modified objective function. Based on the alternating direction method of multipliers (ADMM) framework, we develop an algorithm, called ADMM-MCP, to minimize the modified objective function. The convergent proof of the proposed ADMM-MCP algorithm is also presented. Simulation results show that the proposed ADMM-MCP algorithm is superior to many existing center selection algorithms under the concurrent fault situation.

Aerodynamic Parameters Estimation Using Radial Basis Function Neural Partial Differentiation Method

Aerodynamic parameter estimation involves modelling of force and moment coefficients and computation of stability and control derivatives from recorded flight data. This problem is extensively studied in the past using classical approaches such as output error, filter error and equation error methods. An alternative approach to these model based methods is the machine learning such as artificial neural network. In this paper, radial basis function neural network (RBF NN) is used to model the lateral-directional force and moment coefficients. The RBF NN is trained using k-means clustering algorithm for finding the centers of radial basis function and extended Kalman filter for obtaining the weights in the output layer. Then, a new method is proposed to obtain the stability and control derivatives. The first order partial differentiation is performed analytically on the radial basis function neural network approximated output. The stability and control derivatives are computed at each training data point, thus reducing the post training time and computational efforts compared to hitherto delta method and its variants. The efficacy of the identified model and proposed neural derivative method is demonstrated using real time flight data of ATTAS aircraft. The results from the proposed approach compare well with those from the other.

NNIRSS: neural network-based intelligent routing scheme for SDN

With the increasing diversification of network applications, SDN tends to be inefficient to satisfy the diversified application demands. Meanwhile, the continuous update of OpenFlow and flow table expansion causes the efficiency of routing and forwarding ability decreased as well as the storage space of ternary content addressable memory (TCAM) occupied by flow tables increased. In this paper, we present NNIRSS, a novel neural network (NN)-based intelligent routing scheme for SDN, which leverages the centralized controller to achieve transmission patterns of data flow by utilizing NN and replaces flow table with well-trained NN in the form of NN packet. The route of data flow can be predicted based on its application type to meet the quality of service requirements of network applications. Furthermore, we devise a radial basis function neural network-based intelligent routing mechanism. With combining APC-III and K-means algorithm, we propose APC-K-means algorithm to determine radial basis function centers. Finally, the simulation results demonstrate that our proposed NNIRSS is feasible and effective. It can reduce storage space of TCAM and routing time overhead as well as improve routing efficiency.

Automatic segmentation of cross-sectional coronary arterial images

We present a novel approach to segment coronary cross-sectional images acquired using catheterization imaging techniques, i.e. intra-vascular ultrasound (IVUS) and optical coherence tomography (OCT). The proposed approach combines cross-sectional segmentation with longitudinal tracking in order to tackle various forms of imaging artifacts and to achieve consistent segmentation. A node-weighted directed graph is constructed on two consecutive cross-sectional frames with embedded shape constraints within individual cross-sections or frames and between consecutive frames. The intra-frame constraints are derived from a set of training samples and are embedded in both graph construction and its cost function. The inter-frame constraints are imposed by tracking the borders of interest across multiple frames. The coronary images are transformed from Cartesian coordinates to polar coordinates. Graph partition can then be formulated as searching an optimal interface in the node-weighted directed graph without user initialization. It also allows efficient parametrization of the border using radial basis function (RBF) and thus reduces the tracking of a large number of border points to a very few RBF centers. Moreover, we carry out supervised column-wise tissue classification in order to automatically optimize the feature selection. Instead of empirically assigning weights to different feature detectors, we dynamically and automatically adapt those weighting depending on the tissue compositions in each individual column of pixels. The proposed approach is applied to IVUS and OCT images. Both qualitative and quantitative results show superior performance of the proposed method compared to a number of alternative segmentation techniques.

ADMM-Based Algorithm for Training Fault Tolerant RBF Networks and Selecting Centers.

In the training stage of radial basis function (RBF) networks, we need to select some suitable RBF centers first. However, many existing center selection algorithms were designed for the fault-free situation. This brief develops a fault tolerant algorithm that trains an RBF network and selects the RBF centers simultaneously. We first select all the input vectors from the training set as the RBF centers. Afterward, we define the corresponding fault tolerant objective function. We then add an -norm term into the objective function. As the -norm term is able to force some unimportant weights to zero, center selection can be achieved at the training stage. Since the -norm term is nondifferentiable, we formulate the original problem as a constrained optimization problem. Based on the alternating direction method of multipliers framework, we then develop an algorithm to solve the constrained optimization problem. The convergence proof of the proposed algorithm is provided. Simulation results show that the proposed algorithm is superior to many existing center selection algorithms.

Analysis of Abnormal Intra-QRS Potentials in Signal-Averaged Electrocardiograms Using a Radial Basis Function Neural Network.

Abnormal intra-QRS potentials (AIQPs) are commonly observed in patients at high risk for ventricular tachycardia. We present a method for approximating a measured QRS complex using a non-linear neural network with all radial basis functions having the same smoothness. We extracted the high frequency, but low amplitude intra-QRS potentials using the approximation error to identify possible ventricular tachycardia. With a specified number of neurons, we performed an orthogonal least squares algorithm to determine the center of each Gaussian radial basis function. We found that the AIQP estimation error arising from part of the normal QRS complex could cause clinicians to misjudge patients with ventricular tachycardia. Our results also show that it is possible to correct this misjudgment by combining multiple AIQP parameters estimated using various spread parameters and numbers of neurons. Clinical trials demonstrate that higher AIQP-to-QRS ratios in the X, Y and Z leads are visible in patients with ventricular tachycardia than in normal subjects. A linear combination of 60 AIQP-to-QRS ratios can achieve 100% specificity, 90% sensitivity, and 95.8% total prediction accuracy for diagnosing ventricular tachycardia.