Structural Complexity and Informational Transfer in Spatial Log-Gaussian Cox Processes

Adriana Medialdea,Jorge Mateu,José Miguel Angulo

doi:10.3390/e23091135

Adriana Medialdea, Jorge Mateu + Show 1 more

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https://doi.org/10.3390/e23091135

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Journal: Entropy	Publication Date: Aug 31, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: University of Granada, Jaume I University

Abstract

The doubly stochastic mechanism generating the realizations of spatial log-Gaussian Cox processes is empirically assessed in terms of generalized entropy, divergence and complexity measures. The aim is to characterize the contribution to stochasticity from the two phases involved, in relation to the transfer of information from the intensity field to the resulting point pattern, as well as regarding their marginal random structure. A number of scenarios are explored regarding the Matérn model for the covariance of the underlying log-intensity random field. Sensitivity with respect to varying values of the model parameters, as well as of the deformation parameters involved in the generalized informational measures, is analyzed on the basis of regular lattice partitionings. Both a marginal global assessment based on entropy and complexity measures, and a joint local assessment based on divergence and relative complexity measures, are addressed. A Poisson process and a log-Gaussian Cox process with white noise intensity, the first providing an upper bound for entropy, are considered as reference cases. Differences regarding the transfer of structural information from the intensity field to the subsequently generated point patterns, reflected by entropy, divergence and complexity estimates, are discussed according to the specifications considered. In particular, the magnitude of the decrease in marginal entropy estimates between the intensity random fields and the corresponding point patterns quantitatively discriminates the global effect of the additional source of variability involved in the second phase of the double stochasticity.

Highlights

Log-Gaussian Cox processes define a class of doubly stochastic Poisson processes [1]where the Gaussian intensity-generating function is transformed through exponentiation.These processes allow the generation of point patterns through a stochastic two-step procedure, where the clustering structure observed in the pattern is due to the inclusion of random heterogeneities in the intensity function
The structural properties of random fields and point patterns can be characterized by means of informational and complexity measures
Regarding the local coherence measured in terms of divergence, we can observe an enlargement of the structural information transferred as a result of the increase in the mean and variance parameter values

Summary

Introduction

Log-Gaussian Cox processes define a class of doubly stochastic Poisson processes [1]where the Gaussian intensity-generating function is transformed through exponentiation.These processes (see [2] for a formal definition and properties of log-Gaussian Cox processes) allow the generation of point patterns through a stochastic two-step procedure, where the clustering structure observed in the pattern is due to the inclusion of random heterogeneities in the intensity function. The concept of entropy, first defined in the context of Information Theory by Shannon [11], and generalized by Rényi [12], as the uncertainty contained in a probability distribution, can be used to quantify the degree of inhomogeneity of each phase of the process.

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