  We introduce a family of local inhomogeneous mark-weighted summary
statistics, of order two and higher, for general marked point processes.
Depending on how the involved weight function is specified, these summary
statistics capture different kinds of local dependence structures. We first
derive some basic properties and show how these new statistical tools can be
used to construct most existing summary statistics for (marked) point
processes. We then propose a local test of random labelling. This procedure
allows us to identify points, and consequently regions, where the random
labelling assumption does not hold, e.g.~when the (functional) marks are
spatially dependent. Through a simulation study we show that the test is able
to detect local deviations from random labelling. We also provide an
application to an earthquake point pattern with functional marks given by
seismic waveforms.
