Online identification using radial basis function neural network coupled with KPCA

In this paper, we are going to propose an online radial basis function (RBF) neural network algorithm without any preprocessing step. Then a kernel principal component analysis (KPCA) is coupled with the proposed online RBF neural network algorithm. Indeed, the KPCA method is used as a preprocessing step to reduce the feature dimension which fed to the RBF neural network. Reducing memory requirements of the models makes RBF neural network training efficient and fast. These two proposed algorithms are applied, with success, for identification of a mobile robot position. The simulation results present that the used sigmoid function as a kernel, compared to other kernel functions, which gives an excellent model and a minimum mean square error.     

Scale-based clustering using the radial basis function network

This paper shows how scale-based clustering can be done using the radial basis function network (RBFN), with the RBF width as the scale parameter and a dummy target as the desired output. The technique suggests the "right" scale at which the given data set should be clustered, thereby providing a solution to the problem of determining the number of RBF units and the widths required to get a good network solution. The network compares favorably with other standard techniques on benchmark clustering examples. Properties that are required of non-Gaussian basis functions, if they are to serve in alternative clustering networks, are identified. This work, on the whole, points out an important role played by the width parameter in RBFN, when observed over several scales, and provides a fundamental link to the scale space theory developed in computational vision.     

GPS orbit approximation using radial basis function networks

We present solutions for GPS orbit computation from broadcast and precise ephemerides using a group of artificial neural networks (ANNs), i.e. radial basis function networks (RBFNs). The problem of broadcast orbit correction, resulting from precise ephemerides, has already been solved using traditional polynomial and trigonometric interpolation. As an alternative approach RBFN broadcast orbit correction produces results within the accuracy range of the traditional methods. Our study shows RBFN broadcast orbit correction performs well also near the end of data intervals and for short data spans (～20 min). Regarding limitations of polynomial and trigonometric extrapolation, the most significant advantage of using RBFNs over the traditional methods for GPS broadcast orbit approximation arises from its short time prediction capability.     

Training radial basis function networks using biogeography-based optimizer

Training artificial neural networks is considered as one of the most challenging machine learning problems. This is mainly due to the presence of a large number of solutions and changes in the search space for different datasets. Conventional training techniques mostly suffer from local optima stagnation and degraded convergence, which make them impractical for datasets with many features. The literature shows that stochastic population-based optimization techniques suit this problem better and are reliably alternative because of high local optima avoidance and flexibility. For the first time, this work proposes a new learning mechanism for radial basis function networks based on biogeography-based optimizer as one of the most well-regarded optimizers in the literature. To prove the efficacy of the proposed methodology, it is employed to solve 12 well-known datasets and compared to 11 current training algorithms including gradient-based and stochastic approaches. The paper considers changing the number of neurons and investigating the performance of algorithms on radial basis function networks with different number of parameters as well. A statistical test is also conducted to judge about the significance of the results. The results show that the biogeography-based optimizer trainer is able to substantially outperform the current training algorithms on all datasets in terms of classification accuracy, speed of convergence, and entrapment in local optima. In addition, the comparison of trainers on radial basis function networks with different neurons size reveal that the biogeography-based optimizer trainer is able to train radial basis function networks with different number of structural parameters effectively.

     

Derivative-based global sensitivity measure using radial basis function

Structural and Multidisciplinary Optimization - Global sensitivity analysis (GSA) is always used to measure the contribution of input variables on the variation of model output. In the fields of...     

A sequential multi-category classifier using radial basis function networks

This paper presents a new sequential multi-category classifier using radial basis function (SMC-RBF) network for real-world classification problems. The classification algorithm processes the training data one by one and builds the RBF network starting with zero hidden neuron. The growth criterion uses the misclassification error, the approximation error to the true decision boundary and a distance measure between the current sample and the nearest neuron belonging to the same class. SMC-RBF uses the hinge loss function (instead of the mean square loss function) for a more accurate estimate of the posterior probability. For network parameter updates, a decoupled extended Kalman filter is used to reduce the computational overhead. Performance of the proposed algorithm is evaluated using three benchmark problems, viz., image segmentation, vehicle and glass from the UCI machine learning repository. In addition, performance comparison has also been done on two real-world problems in the areas of remote sensing and bio-informatics. The performance of the proposed SMC-RBF classifier is also compared with the other RBF sequential learning algorithms like MRAN, GAP-RBFN, OS-ELM and the well-known batch classification algorithm SVM. The results indicate that SMC-RBF produces a higher classification accuracy with a more compact network. Also, the study indicates that using a function approximation algorithm for classification problems may not work well when the classes are not well separated and the training data is not uniformly distributed among the classes.     

Sequential approximate multi-objective optimization using radial basis function network

In industrial design optimization, objectives and constraints are generally given as implicit form of the design variables, and are evaluated through computationally intensive numerical simulation. Under this situation, response surface methodology is one of helpful approaches to design optimization. One of these approaches, known as sequential approximate optimization (SAO), has gained its popularity in recent years. In SAO, the sampling strategy for obtaining a highly accurate global minimum remains a critical issue. In this paper, we propose a new sampling strategy using sequential approximate multi-objective optimization (SAMOO) in radial basis function (RBF) network. To identify a part of the pareto-optimal solutions with a small number of function evaluations, our proposed sampling strategy consists of three phases: (1) a pareto-optimal solution of the response surfaces is taken as a new sampling point; (2) new points are added in and around the unexplored region; and (3) other parts of the pareto-optimal solutions are identified using a new function called the pareto-fitness function. The optimal solution of this pareto-fitness function is then taken as a new sampling point. The upshot of this approach is that phases (2) and (3) add sampling points without solving the multi-objective optimization problem. The detailed procedure to construct the pareto-fitness function with the RBF network is described. Through numerical examples, the validity of the proposed sampling strategy is discussed.     

Nonlinear systems parameters estimation using radial basis function network

In this paper, a new on-line scheme for the state and parameter estimation of a large class of nonlinear systems is presented. This scheme uses a radial basis function neuronal predictor with the on-line learning of weights. The algorithms developed are potentially useful for adjusting the controller parameters of variable speed drives. The other interesting feature of the proposed method is its application to failure and fault detection. The parameter identification scheme is an algebraic method combined with state estimation. The asymptotic convergence of the estimates to their nominal values is achieved using the Lyapunov's arguments. The simulation results and the real-time estimation of both rotor resistance and speed of an induction motor based on this approach, show rapidly converging estimates in spite of the measurements noise, discretization effects, parameters uncertainties (e.g. inaccuracies on motor inductance values) and modeling inaccuracies. The other applications of the proposed method include the on-line estimation of the parameters of a synchronous generator.     

Grey-box radial basis function modelling

A fundamental principle in data modelling is to incorporate available a priori information regarding the underlying data generating mechanism into the modelling process. We adopt this principle and consider grey-box radial basis function (RBF) modelling capable of incorporating prior knowledge. Specifically, we show how to explicitly incorporate the two types of prior knowledge: (i) the underlying data generating mechanism exhibits known symmetric property, and (ii) the underlying process obeys a set of given boundary value constraints. The class of efficient orthogonal least squares regression algorithms can readily be applied without any modification to construct parsimonious grey-box RBF models with enhanced generalisation capability.     

Multiplication-free radial basis function network

For the purpose of adaptive function approximation, a new radial basis function network is proposed which is nonlinear in its parameters. The goal is to reduce significantly the computational effort for a serial processor, by avoiding multiplication in both the evaluation of the function model and the computation of the parameter adaptation. The approximation scheme makes use of a grid-based Gaussian basis function network. Due to the local support of digitally implemented Gaussian functions the function representation is parametric local and therefore well suited for an online implementation on a microcomputer. A gradient descent based nonlinear learning algorithm is presented and the convergence of the algorithm is proved.     

Classification and dimensional reduction using restricted radial basis function networks

One of the most efficient means to understand complex data is by visualizing them in two- or three-dimensional space. As meaningful data are likely to be high dimensional, visualizing them requires dimensional reduction algorithms, which objective is to map high-dimensional data into low-dimensional space while preserving some of their underlying structures. For labeled data, their low-dimensional representations should embed their classifiability so that their class-structures become visible. It is also beneficial if an algorithm can classify labeled input while at the same time executes dimensional reduction to visually offer information regarding the data’s structure to give rational behind the classification. However, most of the currently available dimensional reduction methods are not usually equipped with classification features, while most classification algorithm lacks transparencies in rationalizing their decisions. In this paper, the restricted radial basis function networks (rRBF), a recently proposed supervised neural network with low-dimensional internal representation, is utilized for visualizing high-dimensional data while also performing classification. The primary focus of this paper is to empirically explain the classifiability and visual transparency of the rRBF.

     

Blind equalization using a predictive radial basis function neural network

In this paper, we propose a novel blind equalization approach based on radial basis function (RBF) neural networks. By exploiting the short-term predictability of the system input, a RBF neural net is used to predict the inverse filter output. It is shown here that when the prediction error of the RBF neural net is minimized, the coefficients of the inverse system are identical to those of the unknown system. To enhance the identification performance in noisy environments, the improved least square (ILS) method based on the concept of orthogonal distance to reduce the estimation bias caused by additive measurement noise is proposed here to perform the training. The convergence rate of the ILS learning is analyzed, and the asymptotic mean square error (MSE) of the proposed predictive RBF identification method is derived theoretically. Monte Carlo simulations show that the proposed method is effective for blind system identification. The new blind technique is then applied to two practical applications: equalization of real-life radar sea clutter collected at the east coast of Canada and deconvolution of real speech signals. In both cases, the proposed blind equalization technique is found to perform satisfactory even when the channel effects and measurement noise are strong.     

High-speed face recognition using self-adaptive radial basis function neural networks

In this work, we have proposed a self-adaptive radial basis function neural network (RBFNN)-based method for high-speed recognition of human faces. It has been seen that the variations between the images of a person, under varying pose, facial expressions, illumination, etc., are quite high. Therefore, in face recognition problem to achieve high recognition rate, it is necessary to consider the structural information lying within these images in the classification process. In the present study, it has been realized by modeling each of the training images as a hidden layer neuron in the proposed RBFNN. Now, to classify a facial image, a confidence measure has been imposed on the outputs of the hidden layer neurons to reduce the influences of the images belonging to other classes. This process makes the RBFNN as self-adaptive for choosing a subset of the hidden layer neurons, which are in close neighborhood of the input image, to be considered for classifying the input image. The process reduces the computation time at the output layer of the RBFNN by neglecting the ineffective radial basis functions and makes the proposed method to recognize face images in high speed and also in interframe period of video. The performance of the proposed method has been evaluated on the basis of sensitivity and specificity on two popular face recognition databases, the ORL and the UMIST face databases. On the ORL database, the best average sensitivity (recognition) and specificity rates are found to be 97.30 and 99.94%, respectively using five samples per person in the training set. Whereas, on the UMIST database, the above quantities are found to be 96.36 and 99.81%, respectively using eight samples per person in the training set. The experimental results indicate that the proposed method outperforms some of the face recognition approaches.     

Identification of nonlinear discrete-time systems using raised-cosine radial basis function networks

An effective technique for identifying nonlinear discrete-time systems using raised-cosine radial basis function (RBF) networks is presented. Raised-cosine RBF networks are bounded-input bounded-output stable systems, and the network output is a continuously differentiable function of the past input and the past output. The evaluation speed of an n-dimensional raised-cosine RBF network is high because, at each discrete time, at most 2ⁿ RBF terms are nonzero and contribute to the output. As a consequence, raised-cosine RBF networks can be used to identify relatively high-order nonlinear discrete-time systems. Unlike the most commonly used RBFs, the raised-cosine RBF satisfies a constant interpolation property. This makes raised-cosine RBF highly suitable for identifying nonlinear systems that undergo saturation effects. In addition, for the important special case of a linear discrete-time system, a first-order raised-cosine RBF network is exact on the domain over which it is defined, and it is minimal in terms of the number of distinct parameters that must be stored. Several examples, including both physical systems and benchmark systems, are used to illustrate that raised-cosine RBF networks are highly effective in identifying nonlinear discrete-time systems.     

Shape preserving surface reconstruction using locally anisotropic radial basis function interpolants

In this paper we deal with the problem of reconstructing surfaces from unorganized sets of points, while capturing the significant geometry details of the modelled surface, such as edges, flat regions, and corners. This is obtained by exploiting the good approximation capabilities of the radial basis functions (RBF), the local nature of the method proposed in [1], and introducing information on shape features and data anisotropies detected from the given surface points.The result is a shape-preserving reconstruction, given by a weighted combination of locally aniso tropic interpolants. For each local interpolant the anisotropy is obtained by replacing the Euclidean norm with a suitable metric which takes into account the local distribution of the points. Thus hyperellipsoid basis functions, named anisotropic RBFs, are defined. Results from the application of the method to the reconstruction of object surfaces in ℝ³ are presented, confirming the effectiveness of the approach.     

Speech recognition using pulse-coupled neural networks with a radial basis function

We have developed a novel pulse-coupled neural network (PCNN) for speech recognition. One of the advantages of the PCNN is in its biologically based neural dynamic structure using feedback connections. To recall the memorized pattern, a radial basis function (RBF) is incorporated into the proposed PCNN. Simulation results show that the PCNN with a RBF can be useful for phoneme recognition. This work was presented in part at the 7th International Symposium on Artificial Life and Robotics, Oita, Japan, January 16–18, 2002     

Control of chaotic dynamical systems using radial basis function network approximators

This paper presents a general control method based on radial basis function networks (RBFNs) for chaotic dynamical systems. For many chaotic systems that can be decomposed into a sum of a linear and a nonlinear part, under some mild conditions the RBFN can be used to well approximate the nonlinear part of the system dynamics. The resulting system is then dominated by the linear part, with some small or weak residual nonlinearities due to the RBFN approximation errors. Thus, a simple linear state-feedback controller can be devised, to drive the system response to a desirable set-point. In addition to some theoretical analysis, computer simulations on two representative continuous-time chaotic systems (the Duffing and the Lorenz systems) are presented to demonstrate the effectiveness of the proposed method.     

Adaptive control of underwater vehicle-manipulator systems using radial basis function networks

This paper deals with a control scheme for underwater vehicle-manipulator systems with the dynamics of thrusters in the presence of uncertainties in system parameters. We have developed two controllers that overcome thruster nonlinearities, which cause an uncontrollable system: one is a regressor-based adaptive controller and the other is a robust controller. However, the structure of the adaptive controller is very complex due to the feedforward terms including the regressors of dynamic system models, and the error feedback gains of the robust controller with a good control performance are excessively high due to the lack of feedforward terms. In this paper we develop an adaptive controller that uses radial basis function networks instead of the feedforward terms. The replacement leads to a moderately high gain controller whose structure is simpler than that of the regressor-based adaptive controller.     

Cooperative learning for radial basis function networks using particle swarm optimization

This paper presents a new evolutionary cooperative learning scheme, able to solve function approximation and classification problems with improved accuracy and generalization capabilities. The proposed method optimizes the construction of radial basis function (RBF) networks, based on a cooperative particle swarm optimization (CPSO) framework. It allows for using variable-width basis functions, which increase the flexibility of the produced models, while performing full network optimization by concurrently determining the rest of the RBF parameters, namely center locations, synaptic weights and network size. To avoid the excessive number of design variables, which hinders the optimization task, a compact representation scheme is introduced, using two distinct swarms. The first swarm applies the non-symmetric fuzzy means algorithm to calculate the network structure and RBF kernel center coordinates, while the second encodes the basis function widths by introducing a modified neighbor coverage heuristic. The two swarms work together in a cooperative way, by exchanging information towards discovering improved RBF network configurations, whereas a suitably tailored reset operation is incorporated to help avoid stagnation. The superiority of the proposed scheme is illustrated through implementation in a wide range of benchmark problems, and comparison with alternative approaches.     

Robust radial basis function neural networks

Function approximation has been found in many applications. The radial basis function (RBF) network is one approach which has shown a great promise in this sort of problems because of its faster learning capacity. A traditional RBF network takes Gaussian functions as its basis functions and adopts the least-squares criterion as the objective function, However, it still suffers from two major problems. First, it is difficult to use Gaussian functions to approximate constant values. If a function has nearly constant values in some intervals, the RBF network will be found inefficient in approximating these values. Second, when the training patterns incur a large error, the network will interpolate these training patterns incorrectly. In order to cope with these problems, an RBF network is proposed in this paper which is based on sequences of sigmoidal functions and a robust objective function. The former replaces the Gaussian functions as the basis function of the network so that constant-valued functions can be approximated accurately by an RBF network, while the latter is used to restrain the influence of large errors. Compared with traditional RBF networks, the proposed network demonstrates the following advantages: (1) better capability of approximation to underlying functions; (2) faster learning speed; (3) better size of network; (4) high robustness to outliers.