Constraints on pure point diffraction on aperiodic point patterns of finite local complexity

We consider the pure point part of the diffraction on families of aperiodic point sets obeying common local rules. It is shown that imposing such rules results in linear constraints on the partial diffraction amplitudes. These relations can be explicitly derived from the geometry of the prototile space representing the local rules.

Spatial Interactions in Business and Housing Location Models

The paper combines theoretical models of housing and business locations and shows that they have the same determinants. It evidences that classical, behavioural, new economic geography, evolutionary and co-evolutionary frameworks apply simultaneously, and one should consider them jointly when explaining urban structure. We use quantitative tools in a theory-guided factors induction approach to show the complexity of location models. The paper discusses and measures spatial phenomena as distance-decaying gradients, spatial discontinuities, densities, spillovers, spatial interactions, agglomerations, and as multimodal processes. We illustrate the theoretical discussion with an empirical case of interacting point-patterns for business, housing, and population. The analysis reveals strong links between housing valuation and business location and profitability, accompanied by the related spatial phenomena. It also shows that assumptions concerning unimodal spatial urban structure, the existence of rational maximisers, distance-decaying externalities, and a single pattern of behaviour, do not hold. Instead, the reality entails consideration of multimodality, a mixture of maximisers and satisfiers, incomplete information, appearance of spatial interactions, feed-back loops, as well as the existence of persistence of behaviour, with slow and costly adjustments of location.

Generalizing Boltzmann Configurational Entropy to Surfaces, Point Patterns and Landscape Mosaics

Several methods have been recently proposed to calculate configurational entropy, based on Boltzmann entropy. Some of these methods appear to be fully thermodynamically consistent in their application to landscape patch mosaics, but none have been shown to be fully generalizable to all kinds of landscape patterns, such as point patterns, surfaces, and patch mosaics. The goal of this paper is to evaluate if the direct application of the Boltzmann relation is fully generalizable to surfaces, point patterns, and landscape mosaics. I simulated surfaces and point patterns with a fractal neutral model to control their degree of aggregation. I used spatial permutation analysis to produce distributions of microstates and fit functions to predict the distributions of microstates and the shape of the entropy function. The results confirmed that the direct application of the Boltzmann relation is generalizable across surfaces, point patterns, and landscape mosaics, providing a useful general approach to calculating landscape entropy.

Measuring Similarity of Deforestation Patterns in Time and Space across Differences in Resolution

This article demonstrated an easily applicable method for measuring the similarity between a pair of point patterns, which applies to spatial or temporal data sets. Such a measurement was performed using similarity-based pattern analysis as an alternative to conventional approaches, which typically utilize straightforward point-to-point matching. Using our approach, in each point data set, two geometric features (i.e., the distance and angle from the centroid) were calculated and represented as probability density functions (PDFs). The PDF similarity of each geometric feature was measured using nine metrics, with values ranging from zero (very contrasting) to one (exactly the same). The overall similarity was defined as the average of the distance and angle similarities. In terms of sensibility, the method was shown to be capable of measuring, at a human visual sensing level, two pairs of hypothetical patterns, presenting reasonable results. Meanwhile, in terms of the method′s sensitivity to both spatial and temporal displacements from the hypothetical origin, the method is also capable of consistently measuring the similarity of spatial and temporal patterns. The application of the method to assess both spatial and temporal pattern similarities between two deforestation data sets with different resolutions was also discussed.

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

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.

Hierarchical log Gaussian Cox process for regeneration in uneven-aged forests

We propose a hierarchical log Gaussian Cox process (LGCP) for point patterns, where a set of points $$\varvec{x}$$ x affects another set of points $$\varvec{y}$$ y but not vice versa. We use the model to investigate the effect of large trees on the locations of seedlings. In the model, every point in $$\varvec{x}$$ x has a parametric influence kernel or signal, which together form an influence field. Conditionally on the parameters, the influence field acts as a spatial covariate in the intensity of the model, and the intensity itself is a non-linear function of the parameters. Points outside the observation window may affect the influence field inside the window. We propose an edge correction to account for this missing data. The parameters of the model are estimated in a Bayesian framework using Markov chain Monte Carlo where a Laplace approximation is used for the Gaussian field of the LGCP model. The proposed model is used to analyze the effect of large trees on the success of regeneration in uneven-aged forest stands in Finland.