Summary statistics for spatio-temporal point processes on linear networks

We propose novel second/higher-order summary statistics for inhomogeneous spatio-temporal point processes when the spatial locations are limited to a linear network. More specifically, letting the spatial distance between events be measured by a regular distance metric, appropriate forms of $K$- and $J$-functions are introduced, and their theoretical relationships are studied. The theoretical forms of our proposed summary statistics are investigated under homogeneity, Poissonness, and independent thinning. Moreover, non-parametric estimators are derived, facilitating the use of our proposed summary statistics to study the spatio-temporal dependence between events. Through simulation studies, we demonstrate that our proposed $J$-function effectively identifies spatio-temporal clustering, inhibition, and randomness. Finally, we examine spatio-temporal dependencies for street crimes in Valencia, Spain, and traffic accidents in New York, USA.