# Predicting the Compressive Strength of Concrete Using an RBF-ANN Model
## 3 The RBF Artificial Neural Network  
Now we have the function \(\phi\) with 7 arguments:  
\[\eta=\phi(\upvarepsilon) \tag{5}\]  
in which  
\[\upvarepsilon=\left[ \begin{array}{cccc}\upvarepsilon_{1}&\upvarepsilon_{2}& \cdots&\upvarepsilon_{7}\end{array} \right]^{T} \tag{6}\]  
where all the arguments are in the range of 0 to 1. The linear combination of \(n\) RBFs is applied to approximate the input-output relation:  
\[\eta\approx\dot{\eta}=w_{0}+\sum_{j=1}^{n}w_{j}g_{j}(\upvarepsilon) \tag{7}\]  
in which \(g_{j}\) is the \(j\)-th RBF and \(w_{j}\) is its weight while \(w_{0}\) represents the threshold. The larger value of \(n\) indicates the better the RBF-ANN fits the input-output relation in the training data, but it also has a higher risk of over-fitting. With \(N\) pieces of training data, i.e., \(N\) sets of \(\upvarepsilon_{1},\upvarepsilon_{2},\ldots,\upvarepsilon_{7}\) and \(\eta\), a linear-algebraic system can be formed  
\[\mathbf{A}\mathbf{W}=\mathbf{\eta} \tag{8}\]  
in which  
\[\mathbf{A}=\left[ \begin{array}{cccccc}1&g_{1}(\upvarepsilon_{1})&\cdots& \cdots&\cdots&g_{n}(\upvarepsilon_{1})\\ 1&\vdots&\ddots&&&\vdots\\ \vdots&\vdots&&g_{j}(\upvarepsilon_{i})&&\vdots\\ \vdots&\vdots&&&\ddots&\vdots\\ 1&g_{1}(\upvarepsilon_{N})&\cdots&\cdots&\cdots&g_{n}(\upvarepsilon_{N}) \end{array} \right] \tag{9}\]  
\[\mathbf{W}=\left[ \begin{array}{cccccc}w_{0}&w_{1}&\cdots&w_{j}&\cdots&w_{n }\end{array} \right]^{\mathbf{T}} \tag{10}\]  
\[\mathbf{\eta}=\left[ \begin{array}{cccccc}\eta_{1}&\eta_{2}&\cdots&\eta_{i}& \cdots&\eta_{N}\end{array} \right]^{\mathbf{T}} \tag{11}\]  
where \(\upvarepsilon_{i}\) and \(\eta_{i}\) represent the input and output of the \(i\)-th piece of data. Equation (8) is a combination of \(N\) linear equations subjected with \(n+1\) unknowns which are the threshold \(w_{0}\) and the weights \(w_{1}\), \(w_{2}\),..., \(w_{n}\). The least-square-error solution can be obtained promptly by using the following matrix operation. \[\mathbf{W}=\left(\mathbf{A}^{\intercal}\mathbf{A}\right)^{-1}\mathbf{A}^{ \intercal}\mathbf{\eta} \tag{12}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Factors** & **Symbol** & **Maximum** & **Minimum** & **Unit** \\ \hline \multirow{5}{*}{Inputs} & water & \(x_{1}\) & 295 & 116.5 & kg/m\({}^{3}\) \\  & cement & \(x_{2}\) & 643 & 74 & kg/m\({}^{3}\) \\  & fine aggregate (sand) & \(x_{3}\) & 1293 & 30 & kg/m\({}^{3}\) \\  & coarse aggregate (gravel) & \(x_{4}\) & 1226 & 436 & kg/m\({}^{3}\) \\  & blast furnace slag & \(x_{5}\) & 440 & 0 & kg/m\({}^{3}\) \\  & fly ash & \(x_{6}\) & 330 & 0 & kg/m\({}^{3}\) \\  & superplasticizer & \(x_{7}\) & 27.17 & 0 & kg/m\({}^{3}\) \\ \hline Output & 28-day compressive strength & \(y\) & 95.3 & 5 & MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The ranges of the inputs and the output of the raw data [26; 27; 35; 36; 37; 38; 39]. Among many RBFs, the Gaussian RBF is chosen in this study:  
\[g_{j}(\xi)=\exp\Bigl{(}-\|\xi-\zeta_{j}\|^{2}/\sigma_{j}^{2}\Bigr{)} \tag{13}\]  
in which \(\zeta_{j}\) represents the center of the \(j\)th RBF  
\[\zeta_{j}=\left[ \begin{array}{cccc}\zeta_{j1}&\zeta_{j2}&\cdots&\zeta_{j7} \end{array} \right]^{T} \tag{14}\]  
while \(\sigma_{j}\) is the shape parameter and \(\|\xi-\zeta_{j}\|\) denotes the Euclidean distance from \(\xi\) to \(\zeta_{j}\). Ususally, the centers of RBFs and their shape parameters are manually pre-determined. Some may use the k-means approach to set \(n\) cluster centers as the RBF centers. Recently, advanced algorithms like tabu and genetic are employed to determine them [30; 32; 34]. The structure of this RBF-ANN is similar to the one-hidden-layer BP-ANN which has just one output. An RBF can be deemed as a neuron in the hidden layer. The RBF centers are analogous to the synaptic weights from the input layer to the hidden layer. The shape parameters of the RBFs are similar to the biases (or thresholds) of the hidden neurons.