Beyond scalar-valued marks - Statistical analysis of object-valued marked spatial point processes

• Applicant: Dr. Matthias Eckardt
• Subject Area: Statistics and Econometrics
• Term: from 2021 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 467634837

While there has been some progress in the analysis of marked spatial point processes within recent years, the statistical investigation and characterisation of many modern real-world spatial point process scenarios still poses substantial challenges. Many modern, commonly high dimensional, applications increasingly require the investigation of marked spatial point process settings where (i) multiple coincident point locations are reported and (ii) the attributes go far beyond the scalar-valued case. In particular, due to the immense technological progress and increase in storage capacities, researchers are now facing an ever-increasing amount of even more complex spatial point process scenarios, where the points and/ or the marks live on more complex domains. While extensions of spatial point processes to spatio-temporal point process settings, network structures and the sphere have been proposed in recent years, the present literature has so far focussed to a large extend only on extensions for the points but not the marks. In consequence, extensions of spatial point process scenarios where the marks themselves are data objects living on more complicated spaces (i.e. object-valued marks) rather than scalar-valued quantities, remain extremely limited. In particular, there is still a lack of corresponding and computationally efficient extensions of second-order mark characteristics to such non-scalar mark scenarios.The goal of this project is to advance the field of marked spatial point processes with object-valued marks both in terms of theory and in terms of applicable methods for modern real data analysis problems. Different techniques for the structural analysis and quantification of potential planar interdependencies for object-valued marked spatial point processes on different domains will be developed and implemented that go well beyond the present theoretical and methodology treatment of marked spatial point processes. Apart from the introduction of a general theory of object-valued marked spatial point processes, the main focus is put on extensions of second-order mark characteristics including e.g. the mark correlation function or the mark variogram to the present context. Starting with the development of point-wise mark characteristics, characteristics based on metric and semimetric distances related to the proximity between distinct object-valued marks will be introduced. The developed tools will allow to investigate and quantify the pairwise relationships between the point attributes while accounting for the distribution of the points over the planar region or the network structure under study. All developments will be provided as open source implementations and be applied to different sources of spatial data.

• DFG Programme: WBP Position