Cross-spectral properties of a spatial point-lattice process

Abstract

This paper shows how spectral analysis can be used to study a hybrid process involving a spatial point process and a lattice process. Asymptotic distributions of spectral statistics for such processes are derived.

A spatial dependence graph model for multivariate spatial hybrid processes

“…where F i denotes the complex conjugate of F i . We note that in case of complete spatial randomness, as pointed out by Mugglestone (1990) and Kanaan (2000), the bias B of ( 14) is…”

“…That is, for integer values s 1 and s 2 , no distinction between exp(−ı(w p s 1 + w q s 2 )) and exp(−ı((w p + 2kπ)s 1 + (w q + 2kπ)s 2 )) can be made (cf. Renshaw and Ford (1983), Mugglestone (1990) and Kanaan (2000)).…”

“…Taking Kanaan (2000) and Kanaan et al (2008) as fundamental references, a spatial hybrid process is characterised as follows. Let Ξ(s) = (N, X(s)) T denote a bivariate spatial hybrid process with point component N and lattice component X(s), respectively.…”

“…However, although some authors have contributed to the joint analysis of time series and point process in the temporal domain including Brillinger (1994), Halliday et al (1995), Henschel et al (2008) and Rigas (1983), mixtures of spatial point processes and spatial lattice data (so-called spatial hybrids) have not been studied much so far. Exceptions such as Augustin et al (1996), Kanaan (2000), and Kanaan et al (2008) remain restricted to at most the bivariate case considering mixtures of one unmarked point and one lattice component. Inspired by the work on spatial graphical models for multitype and multivariate-marked point processes of Eckardt (2016) and Eckardt and Mateu (2019a) and some developments in the analysis of irregularly-spaced time series presented by Bauwens and Hautsch (2009), Engle and Russell (1998) and Hasbrouck (1991), this paper aims to contribute to the multivariate analysis of spatial data.…”

A review on anisotropy analysis of spatial point patterns

“…He concludes that the smoothed periodogram has better theoretical properties than the one based on the smoothed covariance function. Kanaan (2000) gives a detailed description of an approach where the periodogram is smoothed by a weighted moving average technique. The same technique is used by Mugglestone and Renshaw (1996b) with the modification that the moving average procedure is repeated several times resulting in a smoother surface, similar to what would be obtained by using a Gaussian kernel with bandwidth equal to the number of repetitions of the moving average.…”