  In the past decades, the growing amount of network data has lead to many
novel statistical models. In this paper we consider so called geometric
networks. Typical examples are road networks or other infrastructure networks.
But also the neurons or the blood vessels in a human body can be interpreted as
a geometric network embedded in a three-dimensional space. In all these
applications a network specific metric rather than the Euclidean metric is
usually used, which makes the analyses on network data challenging. We consider
network based point processes and our task is to estimate the intensity (or
density) of the process which allows to detect high- and low- intensity regions
of the underlying stochastic processes. Available routines that tackle this
problem are commonly based on kernel smoothing methods. However, kernel based
estimation in general exhibits some drawbacks such as suffering from boundary
effects and the locality of the smoother. In an Euclidean space, the
disadvantages of kernel methods can be overcome by using penalized spline
smoothing. We here extend penalized spline smoothing towards smooth intensity
estimation on geometric networks and apply the approach to both, simulated and
real world data. The results show that penalized spline based intensity
estimation is numerically efficient and outperforms kernel based methods.
Furthermore, our approach easily allows to incorporate covariates, which allows
to respect the network geometry in a regression model framework.
