Radial Basis Function RBF

The radial basis function has its roots in X-ray diffraction, where it was used as long ago as 1930 to calculate diffraction intensities from three-dimensional molecular structures [90]. The RBF representation is a function of atom-atom distances in a molecule and relates to the diffraction properties of the crystal. 

Many years later, it was found that this characteristic of the descriptor could be used for the correlation of biological activity and three-dimensional structure of molecules. The activity of a compound also depends on the distances between atoms (such as H-bond donors or acceptors) in the molecular structure [91]. Adaptation of the RBF function to biological activity led to the so-called 3D-MoRSE code (3D-Molecule Representation of Structures based on Electron diffraction) [92]. The method of RBF calculation can be simplified in order to derive a descriptor that includes significant information and that can be calculated rapidly  

The properties A of all atoms i,j at a distance R are multiplied and added up to give one component of the RBF for the distance R. The complete RBF is a function of R [93-95]. RBF descriptors show similar characteristics as autocorrelation coefficients. RBFs of different molecules can be compared directly without superimposing the molecular structures. 

Virtual screening or in-silico screening is used for the design of targeted H-braries. AU information about the target or known active compounds can be used to remove unfavorable stractures from the library [95, 97]. 

Once the chemical structures are encoded by an appropriate descriptor set, the similarities of descriptors must be calculated. Descriptor similarities are the basis for a selection of compounds according to diversity or similarity [101]. Diversity selection techniques fall into four classes  

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Models of the form y =f(x) or v =/(x1, x2,..., xm) can be linear or nonlinear they can be formulated as a relatively simple equation or can be implemented as a less evident algorithmic structure, for instance in artificial neural networks (ANN), tree-based methods (CART), local estimations of y by radial basis functions (RBF), k-NN like methods, or splines. This book focuses on linear models of the form... 

An algorithm for computing the decision boundary thus requires the choice of the kernel function frequently chosen are radial basis functions (RBFs). A further input parameter is the priority of the size constraint for used in the optimization problem (Equation 5.38). This constraint is controlled by a parameter that is often denoted by y. A large value of y forces the size of to be small, which can lead to an overht and to a wiggly... 

Afantitis et al. investigated the use of radial basis function (RBF) neural networks for the prediction of Tg [140]. Radial basis functions are real-valued functions, whose value only depends on their distance from an origin. Using the dataset and descriptors described in Cao s work [130] (see above), RBF networks were trained. The best performing network models showed high correlations between predicted and experimental values. Unfortunately the authors do not formally report an RMS error, but a cursory inspection of the reported data in the paper would suggest approximate errors of around 10 K. 

Support Vector Machine (SVM) is a classification and regression method developed by Vapnik.30 In support vector regression (SVR), the input variables are first mapped into a higher dimensional feature space by the use of a kernel function, and then a linear model is constructed in this feature space. The kernel functions often used in SVM include linear, polynomial, radial basis function (RBF), and sigmoid function. The generalization performance of SVM depends on the selection of several internal parameters of the algorithm (C and e), the type of kernel, and the parameters of the kernel.31... 

To determine whether alternative ANN architectures can lead to improved resolution and successful agent detection, Radial Basis Function (RBF) networks [106] were considered for the same problem. RBFs are networks with one hidden layer associated with a specific, analytically known function. Each hidden layer node corresponds to a numerical evaluation of the chosen function at a set of parameters Gaussian waveforms are often the functions of choice in RBFs. The outputs of the nodes are multiplied by weights, summed, and added to a linear combination of the inputs, yielding the network outputs. The unknown parameters (multiplicative weights, means and spreads for the Gaussians, and coefficients for the linear combination of the inputs) are determined by training the RBF network to produce desired outputs for specific inputs. 

One other network that has been used with supervised learning is the radial basis function (RBF) network.f Radial functions are relatively simple in form, and by definition must increase (or decrease) monotonically with the distance from a certain reference point. Gaussian functions are one example of radial functions. In a RBF network, the inputs are fed to a layer of RBFs, which in turn are weighted to produce an output from the network. If the RBFs are allowed to move or to change size, or if there is more than one hidden layer, then the RBF network is non-linear. An RBF network is shown schematically for the case of n inputs and m basis functions in Fig. 3. The generalized regression neural network, a special case of the RBF network, has been used infrequently especially in understanding in vitro-in vivo correlations. 

Multiple concentration fields are used here in an attempt to capture the dominant effects brought about by equations 1 and 2 on the spatially discrete anodic and cathodic areas formed during exposure to 5% NaCl solution. Multiion electrolyte simulation has also been documented [12] using a nested radial basis function (RBF) approach to predict concentration profiles around a rotating disk. Here, the evolution of Mr", [02], [OH ] and [H+] fields around a planar interface is predicted, governed by corrosion rates determined using data obtained from the rotating disc technique. 

These transformations are executed by using so-called kernel functions. The kernel functions can be both linear and nonlinear in nature. The most commonly used kernel function is of the latter type and called the radial basis function (RBF). There are a number of parameters, for example, cost functions and various kernel settings, within the SVM applications that will affect the statistical quality of the derived SVM models. Optimization of those variables may prove to be productive in deriving models with improved performance [97]. The original SVM protocol was designed to separate two classes but has later been extended to also handle multiple classes and continuous data [80]. 

Radial Basis functions (RBF) belong to the class of Artificial Neural Networks (ANNs) and are a popular choice for approximating nonlinear functions. RBF (f> has its output symmetric around an associated centre p. 

The term radial distribution functions shouM not be confused with radial basis functions (RBF), a term introduced by Broomhead and Lowe in 1988 and which represents a type of function used for neural networks employing a hidden layer of radial units [37]. 

Initially, networks were trained from data obtained from the experimental design conditions given in Figure 7.3. These were radial basis function (RBF) networks, multilayer perception (MLP) networks, probabilistic neural networks (PNNs), and generalized regression neural networks (GRNNs), as well... 

In the present study, two ANN methods - the FFBP with the Levenberg-Marquardt algorithm and the radial basis functions (RBF) - were employed to estimate the air pollution parameters measured at a station in Istanbul on chosen episode days, with the focus on the particulate matter. The results were compared with those obtained with the multi-linear regression (MLR) method. 

The recently developed global optimization tool for the Calibration of molecular force fields by Simultaneous Modeling of Simulated data (CoSMoS) [56] uses a metamodeling procedure based on radial basis functions (RBFs). It has been shown in [56] that metamodel-based optimizers particularly suit the quest for quickly finding nearly optimal force-field parameters. The metamodels constmcted by CoSMoS describe functional dependencies between the force-field parameters and the relative deviations of the simulated properties to experimental data so that the minimization task is easier to solve. The RBFs are rational symmetric functions fl) M of the form (x) = (llx l) for x e M. For the present optimization... 

Network analysis Artificial neural network (ANN) and Radial basis function (RBF). 

In 2008 Borah et al. [38] proposed that Neural Network based E-Nose, comprising of an array of four tin-oxide gas sensors, can assist tea quality monitoring during quality grading, principal component analysis (PCA) was used to visualise the different aroma profiles. In addition, K-means and Kohonen s self organising map (SOM) cluster analysis was done, multi layer Perceptron (MLP) network, radial basis function (RBF) network, and constructive probabilistic neural network (CPNN) were used for aroma classification [38]. 

A way to overcome this problem is to generate an approximation of complex analysis code that describes the process accurately, but at a much lower cost. Metamodels offer an approximation in that they provide a model of the model . Clarke et al. (2005) [58] suggested metamodelhng techniques, namely response surface methodology (RSM), radial basis function (RBF), kriging model and multivariate adaptive regression sphnes (MARS) as potentially useful approaches. Computer deterministic experiments have been addressed by Charles et al. (1996) [59], Simpson et al. (1998) [60], CappeUeri et al. (2002) [61] and Aguire et al. (2(X)7) [62],... 

The steps for building a radial basis function (RBF) kernel-based SVM model irsing LibSVM are enumerated here (Fig. 3.6). 

For SVM method, determinants for the kernel function and its parameters are the important steps in the method application. The Radial Basis Function (RBF) (Vapnik, 1995, Tax Duin 1999, Scholkopf et al., 1999) were employed for the kernel ... 

The radial basis function (RBF) network is a two layer network whose output nodes form a linear combination of non-linear basis functions computed by the hidden layer nodes [137-143]. The basis functions in the hidden layer produce a significant nonzero response only when the input falls within a small localized region of the input space (receptive field). In general, the hidden layer nodes use Gaussian response functions, with the position (w) and width (a) used as variables ... 

The ANNs were developed in an attempt to imitate, mathematically, the characteristics of the biological neurons. They are composed by intercoimected artificial neurons responsible for the processing of input-output relationships, these relationships are learned by training the ANN with a set of irqmt-output patterns. The ANNs can be used for different proposes approximation of functions and classification are examples of such applications. The most common types of ANNs used for classification are the feedforward neural networks (FNNs) and the radial basis function (RBF) networks. Probabilistic neural networks (PNNs) are a kind of RBFs that uses a Bayesian decision strategy (Dehghani et al., 2006). 

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