Tuesday, Aug 5: 2:55 PM - 3:20 PM
Invited Paper Session
Music City Center
This paper focuses on improving the analysis and modelling of point processes by addressing the limitations of current methods in handling complex spatial dependencies. In spatial statistics, a primary objective is the estimation of the intensity function, which describes how the expected number of events varies in space according to spatial coordinates and possible available covariates. Traditional methods, such as composite likelihood estimation, are often based on Poisson process assumptions. However, these methods may fail to capture the underlying spatial interactions and dependencies in non-Poissonian point processes, leading to overfitting. Spatial clustering, for example, is often misinterpreted as being driven solely by covariates, rather than a combination of covariate effects and spatial dependence.
The main challenge is to accurately model these spatial dependencies. In composite likelihood estimation, spatial interactions are often neglected, if the dependence structure between points becomes asymptotically negligible. While this may be true for some processes, real-world applications typically involve finite data sets, where dependencies can have a significant impact on the results. In these cases, Poisson-based models distort results by attributing all spatial structures to covariates, instead of recognising the influence of clustering or interactions between points.
To overcome these limitations, the paper introduces advanced methodologies that incorporate second-order statistics to better account for spatial dependencies. Second-order statistics, such as Ripley's K-function or pairwise correlation function, play a crucial role in capturing local interactions within point patterns. By exploiting these statistics, the proposed approach provides a deeper understanding of spatial structures, particularly in non-Poissonian processes where spatial dependence and clustering are central factors.
A key contribution of this work is the incorporation of second-order local synthesis statistics into the modelling process, which provides detailed insight into the local structure of point patterns. Local Indicators of Spatial Association (LISA) functions are used to identify localised deviations from assumed spatial relationships, such as random labelling. These functions serve as diagnostic tools, allowing models to be adapted to account for spatial dependencies and to better distinguish between the effects of covariates and spatial interactions.
In addition, we focus on a new family of weighted, inhomogeneous local summary statistics for (functional) marked point processes. These statistics, which can be second-order or higher, are flexible and able to capture a range of local dependence structures subject to the chosen weight function. This framework allows for the construction of various existing summary statistics, making it a versatile tool for the analysis of marked point processes. Going beyond traditional methods, this approach offers a more refined and accurate model for complex point patterns with spatial dependencies.
In terms of practical application, the effectiveness of the proposed methods is proved through simulation studies. These simulations show that the new techniques can detect local deviations from random labelling and more accurately account for spatial interactions than traditional approaches. Furthermore, we present a real-world application involving earthquake point patterns, in which functional marks (such as seismic waveforms) are used to study the spatial dependence between events. This case study illustrates the practical utility of the methodology in the analysis of complex point processes.
In conclusion, this paper contributes to spatial statistics by offering an improved method for estimating the intensity function of point processes, particularly in cases where interactions and spatial dependencies are significant. By incorporating second-order local statistics and adjusting for spatial dependence, the methodology improves inference on the relationship between covariates and the structure of point processes. The introduction of marks-weighted inhomogeneous local summary statistics further improves the analysis of complex spatial patterns, providing a useful tool for various applications in fields such as ecology, epidemiology and seismology.
Spatial Point process
Local Indicators of Spatial Association (LISA)
Inhomogeneous Mark-Weighted Function
Composite likelihood
spatial dependencies
Random labelling