  Intrinsic location functional is a large class of random locations containing
locations that one may encounter in many cases, e.g., the location of the path
supremum/infimum over a given interval, the first/last hitting time, etc. It
has been shown that this notion is very closely related to stationary
stochastic processes, and can be used to characterize stationarity. In this
paper the author firstly identifies a subclass of intrinsic location functional
and proves that this subclass has a deep relationship to stationary increment
processes. Then we describe intrinsic location functionals using random
partially ordered point sets and piecewise linear functions. It is proved that
each random location in this class corresponds to the location of the maximal
element in a random set over an interval, according to certain partial order.
Moreover, the locations changes in a very specific way when the interval of
interest shifts along the real line. Based on these ideas, a generalization of
intrinsic location functional called "local intrinsic location functional" is
introduced and its relationship with intrinsic location functional is
investigated.
