To produce kernel density estimates (KDE) of point processes in a linear network:
\[\lambda(z)= \sum_{i=1}^{n} \frac{1}{\tau} k(\frac{d_{iz}}{\tau})y_i\]
Using the Quartic function:
\[\lambda(z)= \sum_{i=1}^{n} \frac{1}{\tau}(\frac{3}{\pi}(1-\frac{d_{iz}^2}{\tau^2}))y_i\]
Where,
\(\lambda\)(z) = density at location z;
\(\tau\) is bandwidth linear network distance;
\(k\) is the kernel function, typically a function of the ratio of \(d_{iz}\) to \(\tau\);
\(d_{iz}\) is the linear network distance from event \(i\) to location \(z\).
I wanted to implement a network-based KDE in R based on the algorithm outlined in Xie & Yan (2008).