Some properties of local weighted second-order statistics for spatio-temporal point processes

  • Giada AdelfioEmail author
  • Marianna Siino
  • Jorge Mateu
  • Francisco J. Rodríguez-Cortés
Original Paper


Diagnostics of goodness-of-fit in the theory of point processes are often considered through the transformation of data into residuals as a result of a thinning or a rescaling procedure. We alternatively consider here second-order statistics coming from weighted measures. Motivated by Adelfio and Schoenberg (Ann Inst Stat Math 61(4):929–948, 2009) for the temporal and spatial cases, we consider an extension to the spatio-temporal context in addition to focussing on local characteristics. In particular, our proposed method assesses goodness-of-fit of spatio-temporal models by using local weighted second-order statistics, computed after weighting the contribution of each observed point by the inverse of the conditional intensity function that identifies the process. Weighted second-order statistics directly apply to data without assuming homogeneity nor transforming the data into residuals, eliminating thus the sampling variability due to the use of a transforming procedure. We provide some characterisations and show a number of simulation studies.


K-function Local properties Residual analysis Second-order characteristics Spatio-temporal point patterns 



This paper has been partially supported by the national grant of the Italian Ministry of Education University and Research (MIUR) for the PRIN-2015 program, ‘Complex space-time modelling and functional analysis for probabilistic forecast of seismic events’.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Scienze Economiche, Aziendali e StatisticheUniversità degli Studi di PalermoPalermoItaly
  2. 2.Departament of MathematicsUniversity Jaume ICastellón de la PlanaSpain
  3. 3.Escuela de EstadísticaUniversidad Nacional de ColombiaMedellínColombia

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