Radial Basis Function (RBF) networks [1] are based on a three-layer structure (Figure 1):
- an input layer that spreads out a non distorted input information;
- a hidden layer that includes the RBF kernels;
- an output layer that transforms the hidden layer outputs by a weighted linear combination.
The main object of RBF networks, for establishing structure - activity relationships, consists in deriving the best approximation of the function relating molecular descriptors and activity values. This task is achieved by associating a Gaussian function with each node of the hidden layer of the network. The latter evolves by modifying the centroids and the standard deviations of the Gaussian functions. A specific weight is associated with each node in order to strengthen or weaken its contribution.
Figure 1. Three-layer structure defining a RBF network, where wi represent the connection weights.
The number of nodes employed plays a key role to establish effective networks and, then, robust models. The main aim is to use enough neural nodes to obtain good performances but keeping a general behavior. To achieve this object, the nodes are added to the network in a dynamical way, with help of an activation radius. More particularly, a node is activated for a given input point if the distance between the latter and the gaussian centroid is lower than the activation radius.
The network development is based on an iterative process:
- a given compound, represented by a vector in the descriptor hyperspace, is propagated through the network up to the output layer;
- the error between the experimental and computed activity is computed. If this error is higher than a threshold value, the network must be modified;
- if a neural node can be activated, the change consists in modifying the related Gaussian function and its weight,
else a new node is added to the network, in the same position occupied by the input compound;
- a new compounds is added to the network and the steps 1-3 are repeated for all data sets compounds.
The RBF architecture is highly dependent on several parameters that establish the initial number of neural nodes and their position, the standard deviation of the Gaussian functions, the maximal number of nodes, etc. Then, a new network is established for each set of parameters and the software gives the possibility to automatically test all these values inside predefined ranges. To derive the best QSAR models amidst all RBF networks generated, an index based on cross-validation procedures is included in the software.
References
1. Powell M.J.D., The theory of radial basis function approximation, Advances in numerical analysis, 2, 1990, 105-210.
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