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\begin{document}

\section{Radial-Basis Function Networks}
A diagram of a radial-basis function~(RBF) network is shown in Figure~\ref{fig:grbf_ip}.
\begin{figure}
\vspace{3in}
\hskip.75in \special{bmp:c:/mybook/grbf_ip.bmp x=4in y=3in}
\caption{A radial-basis function~(RGF) network.}
\label{fig:grbf_ip}
\end{figure}
The functions $\phi\left(\left\|\b x-\b x^{(i)} \right\| \right)$, $i=1,2,\ldots,N$ are called the radial-basis functions, where $\|\cdot \|$ denotes a $p$-norm that is usually the Euclidean 2-norm. The real paramters $w_i$, $i=1,2, \ldots, N$ are the weights. A common choice for $\phi$ is
\begin{equation}
\phi\left(\left\|\b x-\b x^{(i)} \right\| \right)=\exp\left(-\frac{\left\| \b x-\b x^{(i)}\right\|}{2\sigma_i^2} \right),
\label{eq:grbfunct}
\end{equation}
where $\b x^{(i)}$ is the center of the basis function and $\sigma_i$ is sometimes called the radius of the radial basis function. The function given by~(\ref{eq:grbfunct}) is called the Gaussian radial basis function.

The RBF networks are used to perform interpolation. The interpolation problem can be formulated as:\\
For a given set of $N$ different points,
\[
\left\{ \b x^{(i)}:\b x^{(i)}\in \mb{\Bb R}^n, i=1,2,\ldots, N\right\}
\]
and a corresponding set of $N$ real numbers
\[
\left\{ d_i:d_i\in \mb{\Bb R}, i=1,2,\ldots, N\right\}
\]
find a function
\[
F:\mb{\Bb R}^n \to \mb{\Bb R}
\]
such that
\begin{equation}
F\left( \b x^{(i)}\right)=d_i, i=1,2,\ldots, N.
\label{eq:interp_con}
\end{equation}
The function that the RBF network implements is
\begin{equation}
F(\b x)=\sum_{i=1}^{N}w_i \phi\left(\left\|\b x-\b x^{(i)} \right\| \right).
\label{eq:rbffunimp}
\end{equation}
Substituting the interpolation conditions given by~(\ref{eq:interp_con}) into~(\ref{eq:rbffunimp}) gives
\begin{equation}
\left[\begin{array}{cccc}
\phi_{11} & \phi_{12} & \cdots & \phi_{1N}\\
\phi_{21} & \phi_{22} & \cdots & \phi_{2N}\\
\vdots & \vdots & \ddots & \vdots\\
\phi_{11} & \phi_{12} & \cdots & \phi_{1N}
\end{array}\right]\left[\begin{array}{c}
w_1\\
w_2\\
\vdots\\
w_N
\end{array}\right]=\left[\begin{array}{c}
d_1\\
d_2\\
\vdots\\
d_N
\end{array}\right].
\label{eq:matvecinter}
\end{equation}
Let $\b w=\left[\begin{array}{cccc}
w_1& w_2 &\cdots &w_N
\end{array}\right]$, $\b d=\left[\begin{array}{cccc}
d_1& d_2 &\cdots &d_N
\end{array}\right]$, and
\[
\b{\Phi}=\left[\begin{array}{cccc}
\phi_{11} & \phi_{12} & \cdots & \phi_{1N}\\
\phi_{21} & \phi_{22} & \cdots & \phi_{2N}\\
\vdots & \vdots & \ddots & \vdots\\
\phi_{11} & \phi_{12} & \cdots & \phi_{1N}
\end{array}\right].
\]
Then, using the above notation, we represent~(\ref{eq:matvecinter}) as
\begin{equation}
\b{\Phi}\b w=\b d.
\label{eq:matvecip}
\end{equation}
The matrix $\b{\Phi}$ is called the interpolation matrix. If the interpolation matrix is nonsingular, then we can solve~(\ref{eq:matvecip}) for $\b w$ to obtain
\begin{equation}
\b w=\b{\Phi}^{-1}\b d.
\label{eq:weqqinvd}
\end{equation}
\begin{example}\label{ex:iterusrgf}\rm
We construct an RBF network for the interpolation data given in Table~\ref{tab:interdata}.
\begin{table}
\caption{Interpolation data for Example~\ref{ex:iterusrgf}.}
\begin{center}
\begin{tabular}{cc}
$x^{(i)}$ & $d_i$\\\hline
0 & 0.8\\
6 & 0.2\\
7 & 0.4\\
8 & 0.6\\
10 & 0.3\\\hline
\end{tabular}
\end{center}
\label{tab:interdata}
\end{table}
We set $\sigma_i=0$, $i=1\ldots,5$ and evaluate~(\ref{eq:weqqinvd}) to obtain
\[
\b w=\left[\begin{array}{ccccc}
0.8000& 0.1459& -0.0390& 0.5737& 0.2227
\end{array}\right]^T
\]
The interpolation result is depicted in Figure~\ref{fig:grbf_eip}.
\begin{figure}
\vspace{4.5in}
\special{bmp:c:/mybook/grbf_eip.bmp x=6in y=4.5in}
\caption{Results of interpolating using radial-basis functions.}
\label{fig:grbf_eip}
\end{figure}
In Figure~(\ref{fig:grbf_eip})(a), we used solid line to show the result of interpolating using the Gaussian radial-radial basis functions, while broken line indicates the result of straight line interpolation. The plots of the basis functions that contribute to the interpolation are shown in Figure~(\ref{fig:grbf_eip})(b).
\end{example}
\end{document}