# Prediction of Blade Tip Timing Sensor Waveforms Based on Radial Basis Function Neural Network
## 3 Neural Network Model
### Structure of Neural Network
#### 3.1.1 RBF Neural Network Model  
A radial basis function neural network (RBF NN) is a three-layer network widely used for function approximation and model classification. In 1988, [PERSON], [PERSON], [PERSON] and [PERSON] first applied radial basis functions to neural network design [32].  
The Radial Basis Function (RBF) is one that takes a value that depends only on the distance from the origin of the real-valued function of the method, that is, \(\Phi(x,c)=\Phi(\parallel x-c\parallel)\) (\(\parallel x-c\parallel\) denotes the modulus of the difference vector, or two parameters). Either function \(\Phi\) that satisfies the properties of \(\Phi(x)=\Phi(\parallel x\parallel)\) is called a radial basis function. The most common RBF is a Gaussian kernel function of the form \(k(\parallel x-x_{c}\parallel)=e^{\frac{-\parallel x-x_{c}\parallel}{2\sigma^{2 }}}\), where \(x_{c}\) is the center of the kernel function, and \(\sigma\) is the width parameter of the function, which controls the radial range of action of the function.  
Assume that the N-dimensional space has p input vectors (\(p=1\), \(2\), \(3\),..., \(P\)) whose corresponding target values in the output space are \(d^{p}\), \(p=1\), \(2\), \(3\),..., \(P\), \(p\) pairs of input-output samples constitute the training sample set. Interpretation aims to find a nonlinear mapping function \(F(x)\) such that it satisfies  
\[F(x)=d^{p}\ p=1,\ 2,\ 3,\ldots,P \tag{3}\]  
where \(F\) depicts the interpolating surface.  
The interpolation problem is solved using the RBF technique by selecting \(P\) basis functions to train the data, with individual basis functions of the following form:  
\[\varphi(\parallel x-x^{p}\parallel)\ p=1,\ 2,\ 3,\ldots,P \tag{4}\]  
where the basis function \(\varphi\) is a nonlinear function and the training data point \(x^{p}\) is the center of \(x\). The fundamental function takes the distance between the point \(x\) and the center \(x^{p}\) in the input space as the independent variable of the function. Since the distances are radially homogeneous, the function is called a radial basis function. The definition of the difference function based on the RBF technique is a linear combination of cardinal functions as follows:  
\[F(x)=\sum_{p=1}^{P}\omega_{p}\varphi(\parallel x-x^{p}\parallel) \tag{5}\]  
Figure 2: MP model.  
Taking the interpolation condition of Equation (3) into Equation (5) results in a system of \(P\) linear equations with respect to the unknown coefficient \(\omega^{p}\), (\(p\) = 1, 2, 3, \(\ldots\), \(P\)):  
\[\begin{array}{l}\sum\limits_{p=1}^{P}\omega_{p}\varphi\big{(}\parallel x^{1}- x^{p}\parallel\big{)}=d^{1}\\ \sum\limits_{p=1}^{P}\omega_{p}\varphi\big{(}\parallel x^{2}-x^{p}\parallel \big{)}=d^{2}\\ \vdots\\ \sum\limits_{p=1}^{P}\omega_{p}\varphi\big{(}\parallel x^{p}-x^{p}\parallel \big{)}=d^{P}\end{array} \tag{6}\]  
Let \(\varphi_{ip}=\varphi\big{(}x^{i}-x^{p}\big{)}\), \(i\) = 1, 2, 3, \(\ldots\), \(P\), \(p\) = 1, 2, 3, \(\ldots\), \(P\), replace the above system of equations as follows:  
\[\begin{array}{l}\left[\begin{array}{c}\varphi_{11}\varphi_{12}\cdots \varphi_{1P}\\ \varphi_{21}\varphi_{22}\cdots\varphi_{2P}\\ \cdots,\cdots,\ddots,\ldots\\ \varphi_{P1}\varphi_{P2}\cdots\varphi_{PP}\end{array}\right]\left[\begin{array} []{c}\omega_{1}\\ \omega_{2}\\ \cdots\\ \omega_{P}\end{array}\right]=\left[\begin{array}{c}d^{1}\\ d^{2}\\ \ldots\\ d^{P}\end{array}\right]\end{array} \tag{7}\]  
Suppose \(\Phi\) denotes the \(P\times P\) order matrix of element \(\varphi_{ip}\), where W and d denote the coefficient vector and the desired output vector, respectively, then Equation (7) can be expressed in vector form as follows:  
\[\Phi W=d \tag{8}\]  
where \(\Phi\) is referred to as the interpolation matrix, and if \(\Phi\) is an invertible matrix, then the coefficient vector can be derived from (8), that is:  
\[W=\Phi^{-1}d \tag{9}\]